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Physical Optics
Lecture 1: Wave optics
2017-04-05
Herbert Gross
Physical Optics: Content
2
No Date Subject Ref Detailed Content
1 05.04. Wave optics G Complex fields, wave equation, k-vectors, interference, light propagation,
interferometry
2 12.04. Diffraction B Slit, grating, diffraction integral, diffraction in optical systems, point spread
function, aberrations
3 19.04. Fourier optics B Plane wave expansion, resolution, image formation, transfer function,
phase imaging
4 26.04. Quality criteria and
resolution B
Rayleigh and Marechal criteria, Strehl ratio, coherence effects, two-point
resolution, criteria, contrast, axial resolution, CTF
5 03.05. Polarization G Introduction, Jones formalism, Fresnel formulas, birefringence,
components
6 10.05. Photon optics D Energy, momentum, time-energy uncertainty, photon statistics,
fluorescence, Jablonski diagram, lifetime, quantum yield, FRET
7 17.05. Coherence G Temporal and spatial coherence, Young setup, propagation of coherence,
speckle, OCT-principle
8 24.05. Laser B Atomic transitions, principle, resonators, modes, laser types, Q-switch,
pulses, power
9 31.05. Gaussian beams D Basic description, propagation through optical systems, aberrations
10 07.06. Generalized beams D Laguerre-Gaussian beams, phase singularities, Bessel beams, Airy
beams, applications in superresolution microscopy
11 14.06. PSF engineering G Apodization, superresolution, extended depth of focus, particle trapping,
confocal PSF
12 21.06. Nonlinear optics D Basics of nonlinear optics, optical susceptibility, 2nd and 3rd order effects,
CARS microscopy, 2 photon imaging
13 28.06. Scattering D Introduction, surface scattering in systems, volume scattering models,
calculation schemes, tissue models, Mie Scattering
14 05.07. Miscellaneous G Coatings, diffractive optics, fibers
D = Dienerowitz B = Böhme G = Gross
Complex fields, k-vectors and plane waves
Wave equation
Light Propagation
Interference
Interferometry
3
Contents
Quantities: E electrical field t time
H magnetic field e dielectricity
j current m permeability
r charge density c speed of light
Maxwell equations:
current
induction
source free magnetic
charge
continuity of charges
Basic equations for electromagnetism and optics
4
Maxwell Equations
ree Ediv r
0
00 Hdiv r
t
EjHrot r
0ee
t
HErot r
0
tjdiv
r
Alternative formulation of the Maxwell equations in a medium: M magnitization B magnetic induction D electric displacement
The electromagnetic fields form a transverse wave k: wave vector indicates the direction of propagation
Wavelength (scalar)
Maxwell Equations
E
H
k
0
Bk
iDk
BEk
jiDHk
r
EJ
Jk
MHB
PED
r
r
r
ee
0
0
o
o
nkn
ck
2
2 2
o
c c
k
5
Speed of light in medium
in vacuum
constants
Energy density of a field
Local flow of energy:
Poynting vector
Intensity
6
Energy Density and Poynting vector
n
c
nc
rr
0
0000
11
eee
2
0
2
0
2
02
1EHEu
ee
2
0 EecHES k
e
2
002
1EcI e
8
0 0
12.99792458 10o
mc
se
.104
,10...85.8
70
100
Am
Vs
Vm
As
e
Speed of light in medium
Wavelength in medium
Wave vector in medium
Refraction
7
Fields in Dielectric Media
0cc
n
0
n
0k k n
n
Ref: Saleh / Teich
Description of electromagnetic fields:
- vectorial nature of field strength
- decomposition of the field into components
Propagation plane wave:
- field vector rotates
- projection components are oscillating sinusoidal
yyxx etAetAE )cos(cos
z
x
y
Electromagnetic Fields
8
1. Linear components in phase
2. circular phase difference of 90° between components
3. elliptical arbitrary but constant phase difference
x
y
z
E
E
x
y
z
EE
x
y
z
E
E
Basic Forms of Polarisation
9
10
Basic Wave Optics
Scalar wave
phase function
Phase surface:
- fixed phase for one time
- phase surface perpendicular to
unit vektor e
( , )( ) ( ) e i rA r a r
( , )r
A0
2k r r e const
Ref: W. Osten
Wave harmonic in space and time
Representation with complex
exponential function
Coupling of time and space propagation/development
with
Harmonic Wave
Re{F(t)}
t
T
Re{U(z)}
z
t'
Re{F(t)}
Im{F(t)}
Re{F(t)}
Im{F(t)}
t
t'
,E r t U r F t
tetF ti cosRe
zkezUzik
0cosRe 0
20
ck
( )( , ) i k r tE r t e
11
12
Plane Wave
Condition of a plane wave
y
k r = const.
x
z
k
.constrk
znzxU
2cos~,
cossin
2cos~, zxnzxU
13
Plane and Spherical Waves
Plane wave
wave vector k
Spherical wave
)(),( trkiAetrE
)(),( trkier
AtrE
Ref.: B. Dörband
14
Spherical Wave
Field of a spherical wave
radius in spherical coordinates
( )
( , )i kr te
E r tr
222 zyxr
r
Basis: 1. superposition of solutions
2. separability of coordinates
Plane wave
Spectral expansion
A(k): plane wave spectrum
Dispersion relation: Ewald sphere
Transverse expansion
Main idea:
- Field decomposition in plane waves
- Switch into Fourier space of spatial frequencies
- Propagation of plane wave as simple phase factor
- Back transform into spatial domain
- Superposition of plane wave with modified phase
Plane Wave Expansion
kdekArE rki
)(
2
13
trkietrE
,
2
2222
o
zyxc
nkkkk
E x y z A k k z e dk dkx y
i xk yk
x y
x y, , ( , , )
1
22
k k k k k kT z x y z
A k k z E x y z e dxdyx y
i xk ykx y, , ( , , )
15
Propagation of plane waves:
pure phase factor
1. exact sphere
2. Fresnel quadratic approximation
Evanescent waves
components damped in z
important only for near field setups
Propagation algorithm
x-y-sections are coupled
Paraxial approximation
x-y-section decoupled
Plane Wave Expansion
A k k z A k k A k k ex y x y
ik z
x y
ik zk
k
k
kz
x x
, , , , , ,
0 01
2 2
22
22
0,,
0,,,, 2
yx
yx
vvzizik
yx
kkk
iz
zik
yxyx
eevvA
eekkAzkkA
evanescentkkforkkkik yxyxz ,0222
0
22
),(ˆˆ
),(ˆˆ)','(
222/121
1
yxEFeF
yxEFeFyxE
xy
vviz
xy
xy
zik
xy
yx
z
),(ˆˆ)','(22
1 yxEFeFyxE xy
vvzi
xyyx
16
Propagation by Plane / Spherical Waves
Expansion field in simple-to-propagate waves
1. Spherical waves 2. Plane waves
Huygens principle spectral representation
Fourier approach
rdrErr
erE
rrik
2
'
)('
)'(
x
x'
z
E(x)
eikr
r
)(ˆˆ)'( 1 rEFeFrE xy
zik
xyz
x
x'
z
E(x)
eik z z
17
18
Propagation of Plane Waves
Phase of a plane wave
The spectral component is simply multiplied by a phase
factor in during propagation
the function h is the phase function
Back-transforming this into the spatial domain:
Propagation corresponds to a convolution
with the impulse response function
Fresnel approximation for propagation:
zz z0 1
x
z
zheeee yx
nzi
zinzi
iyx
z
;,
222
22
cos2
2
1 0 0, , , , , ; , ,zi z
x y x y x y x yE z E z e h z E z
...2
11 222
222
yxyx
00
2
002
1200
20
20
22
0;,1
;, dydxeeyxUezi
zyxUyyxxi
zyxi
zyx
zzi
P
1 0, , , ; , ,E x y z H x y z E x y z
yx
yxi
yx ddezhzyxH yx2;,;,
19
Angular Spectrum
Plane wave
Wave number
Spatial frequence: re-scaling of k
Fourier transform to get the plane wave spectrum
z
x
n/
x
z
E( , , ) ( , , ) x y zi k x k y k z
x y z A x y z e
2222 nkkkk zyx
k
2
2( , , ) x y zi x y z
E x y z A e
222
yxzn
2, , , , x y zi x y z
x y z x y zE x y z E e d d d
20
Ewald Sphere
grating
koutkobj
kin
kin
kobj
kout
Ewald sphere
2objk
L
Assuming an object as grating with period L
Scattering of a wave at the object with
- conservation of energy
- conservation of momentum
The outgoing k-vector must be on a sphere:
Ewald's sphere for possible scattered wave vectors
in outk k
in obj outk k k
21
Refraction in k-Space
Plane wave refracted at a plane interface
k-vector should be constant in length
Total internal reflection:
possible if n1 > n2
n2
n1
ky
k1z
k2z
k1y
k1z
k2y
k1y
2
1
1
kz
n2
n1
kz
1. case :
refracted ray
2. case :
beginning
total internal
reflection
3. case :
total internal
reflection
3.
2.
1.
ky
2
1
2
1
2
1
2
1 knkkk zyx
2
2
2
2
2
2
2
2 knkkk zyx
trki
o eEE
22
Evanescent Waves
Usual case in optics:
- refractive index n real
- no damping or loss during light propagation
- spatial frequency real
General case: n and k complex
- damped or evanescent wave
- absorption of the field along the
propagation path
- absorption constant
Damping along z:
- complex refractive index
- absorption constant
- Lambert-Beer law
222
yxzn
, , x yi k x k y zE x y z A e e
222 2
nkk yx
inn 1~
0
0
42
nnk
zeIzI 0)(
1
z
I / I0
1 /
1 / e = 0.368
Metals
Complex refractive index
Alternative formulation:
attenuation
Relation with conductivity
Typical data of some metals
ir ninn ~
inn 1~
e
e
24~ iin
material n k in [m]
gold 0.402 2.54 0.034
silver 0.129 3.25 0.026
aluminum 0.912 6.56 0.013
tungsten 3.50 2.72 0.032
platinum 2.10 3.67 0.023
silicon 4.12 0.048 1.787
23
24
3D Transfer Function - Missing Cone
x
z
missing
cone
transfer function
object
x
z
illumination
scattered
wave
transfer function
object
i
s
obs
Realistic case:
finite numerical aperture
Blue cone:
possible incoming wave direction due
to illumination cone
3D coherent transfer function:
limited green area, that fulfills all
conditions
Missing cone:
certain range of spatial axial spatial
frequencies can not be seen in the image
25
McCutchen Formula and Axial Resolution
n/
P(z)
zz
Dz
x
x
P(x)
Dx
light
cone
Ewald
sphere
transverse
pupil
axial
pupil
cap
nR
2 2 2 2
sin / /x
xv R n NA n NA
D
Imaging of a plane wave at a volume object
x: minimum value resolution
D: maximum interval
Uncertainty relation: D x = 1
Radius of the Ewald sphere
generalized 3D pupil: red area
Transverse resolution due to Abbe
Axial resolution:
- height of the cap of the cone
- McCutchen formula
222
2
2
sin11/
1
cos
11
NA
n
NAnn
nRRvz
z
D
26
3D Transfer Function
z
x
Ewald
sphereforward
2n/
i
i
s
backward
o-max
obj
obj = s - i
Imaging as 3D scattering phenomen
Only special spatial frequencies are allowed due to energy conservation and
momentum preservation
Green circle: supported spatial
frequencies of the transmitted
wave vector
Maxwell equation for the field E, vectorial
The spatial inhomogeneities couples the field
components
Homogeneous, without charges, non-conductive
separation of vector components, scalar
Time independence:
Wave equation of Helmholtz
In coordinate representation
Wave number in medium refractive index n
Fast z-oscillation separated
Slowly varying envelope approximation
2
22
~~
t
EE
e
tierEtrE )(),(
02 D EkE
o
o
nkn
ck
2
ikzezyxEzyxE ),,(),,(
Wave Equation
0ln2
22 rroo E
t
EE e
ee
0),,(
2
22
2
2
2
2
2
2
Ec
zyxn
z
E
y
E
x
E
2
2
2 22
0
2
22E
zik
E
zE k
n x y
nn Eo
( , )
27
Paraxial approximation
paraxial wave equation
Conditions for scalar approximation:
1. Decoupling of field components,
wavelength small in comparison to free diameter
2. No large angles due to geometry,
Computation of field in large distances z
Scalar Helmholtz equation
a
z
Wave Equation
022
2
2
2
z
Eki
y
E
x
E
2
2
E
zk
E
z
0)(2 D rEnko
k
28
1. Spectral representation or orthogonal expansions:
Plan wave expansion, Fourier method
Expansion into gaussian beams in paraxial systems
Expansion into spherical waves, scattering geometries
Expansion into Eigenmodes for boundary problems,
e.g. in fibers, integrated optics, waveguides
2. Integral representations (field on surfaces)
Kirchhoff diffraction integral, Rayleigh-Sommerfeld I+II
Special approximations: Fresnel-, Collins-, Fraunhofer integral
Debye integral
Richards-Wolf vectorial representation
Boundary edge wave approximation
Method of stationary phase, Saddelpoint method
3. Direct solution of the wave equation (volume solution)
Finite difference method
Finite element method
Radial basis functions
Potential methods
Method of lines
Coherent Numerical Field Propagation
29
Solution Methods of the Maxwell Equations
Maxwell-
equations
diffraction
integrals
asymptotic
approximation
Fresnel
approximation
Fraunhofer
approximation
finite
elements
finite
differences
exact/
numerical1st
approximation
direct
solutions of
the PDE
spectral
methods
plane wave
spectrum
vector
potentials
2nd
approximation
finite element
method
boundary
element
method
hybrid method
BEM + FEM
Debye
approximation
Kirchhoff-
integral
Rayleigh-
Sommerfeld
1st kind
Rayleigh-
Sommerfeld
2nd kind
mode
expansion
boundary
edge wave
30
Method Calculation Properties / Applications
Kirchhoff
diffraction integral E r
iE r
e
r rdF
i k r r
FAP
( ) ( ' )'
'
Small Fresnel numbers,
Numerical computation slow
Fourier method of
plane waves )(ˆˆ)'(21 xEFeFxE zvi
II
Large Fresnel numbers
Fast algorithm
Split step beam
propagation
Wave equation: derivatives approximated
En
yxnkE
z
Eik
z
E
o
1
),(2
2
222
2
2
Near field
Complex boundary geometries
Nonlinear effects
Raytracing Ray line law of refraction
r r sj j j j 1 sin '
'sini
n
ni
System components with a aberrations
Materials with index profile
Coherent mode
expansion
Field expansion into modes
n
nn xcxE )()( dxxxEc nn )()( *
Smooth intensity profiles
Fibers and waveguides
Incoherent mode
expansion
Intensity expansion into coherent modes
n
nn xcxI2
)()(
Partial coherent sources
Wave Optical Coherent Beam Propagation
31
Typical change of the intensity profile
Normalized coordinates
Diffraction integral
32
Fresnel Diffraction
z
geometrical
focus
f
a
z
stop
far zone
geometrical
phase
intensity
a
rr
f
avz
f
au
r
;
2;
22
1
0
20
/
2
0
2 22
)(2
),( rrr
r
devJef
EiavuE
ui
uafi
Split Step Propagators
Calculating the field after every small Dz in many single steps
Algorithms: finite differences of plane waves
The field is known in the complete volume
This time-consuming technique is necessary for non-homogeneous media
z
ys
y
Split-step-propagator: 1. Finite Differences
2. Plane wave expansion
many steps
starting
plane
final plane
inhomogeneous
medium
33
Classical beam propagation: split-step approach
Wave equation
two operators: interfaces (material)
and diffraction
Fourier transform
Plane wave decomposition
)(ˆ, zEFzkE x
0
2
2,,
k
zik
xx
x
ezkEzzkE
D
D
zzkEFzzxE x DD ,ˆ),(' 1
zSezzxEzzxE DDD ˆ
,',
EHED
En
yxn
i
kE
ikz
E
o
ˆˆ
1),(
22
12
22
Eik
ED 2
2
1ˆ
En
yxn
i
kES
o
1
),(
2ˆ
2
2
Split Step Propagators
34
Finite Difference Propagation
Paraxial Helmholtz wave equation
Approximation of derivatives
Implicite scheme, linear system of
equations for one steps Dz
Large Fresnelnumber of one voxel,
Fast evaluation of tridiagonal system
Phase nearly flat, no large tilt
Stability:
1. Explicite schemes, usually unstable
2. Good stable implicite schemes exists
z
lateral x
zn+1
xj x
j+1x
j-1
zn
propagation
x
EE
x
E jj
D
2
11
2
11
2
2 2
x
EEE
x
E jjj
D
Enzxnkx
E
z
Enik oooo 222
2
2
),(2
njnjnj
njnjnj
aEcEaE
aEbEaE
,1,,1
1,11,1,1
az
x
D
D2 2 bz
x
zn n ik nj m o o o
D
D
D2 1
2 2
22,
cz
x
zn n ik nj m o o o
D
D
D2
2 2
22,
35
Focussed gaussian beam with spherical aberration
- asymmety intra- vs. extra focal
- sign of spherical aberration has influence
- broadening of beam waist diameter
- diffraction fringes
Gaussian beam propagation in a parabolic gradient
index medium
Refocussing effects
Beam Propagation Examples
c9 = -0.25
c9 = 0.25
c9 = 0
36
37
Intensity of Superposed Fields
CCD is not able to detect phase due to time averaging
Measuring of intensity with simple detector
Measured intensity is time average
Interferometry and holography:
coding of phase information into measurable intensity variation
Contrast / visibility:
normalized difference of two different intensities
(typically maximum / minimum values)
Value between 0...1
General case of two-wave interference
Ref: W. Osten
22
0
1
2r ot
I P E Ae e
max min
max min
I IC
I I
1 2 121 cosI I I C
38
Interference of Waves
The main property is the phase difference
between two waves
Interference of two waves
special case of equal intensites
Maxima of intensity at even phase differences
Minima of intensity at odd phase differences
Interference of plane waves
Interference of spherical waves:
1. outgoing waves
rotational hyperboloids
2. one outgoing, one incoming wave
rotational ellipsoids Ref: W. Osten
jk j k
1 2 1 2 122 cosI I I I I
2jk N
(2 1)jk N
0 122 1 cosI I
1 2k r k r
1 2k r r
1 2k r r
Two beam interference of two waves:
- propagation in the same direction
- same polarization
- phase difference smaller than axial length of coherence
Coherent superposition of waves
Difference of phase / path difference
Number of fringes
location of same phase
Conrast
122121
2
21
cos2 D
IIII
EEI
122
DDs
sN
D
D
2
12
21
21
minmax
minmax2
II
II
II
IIK
Two Beam Interference
39
40
Interference of Two Waves
Superposition of two plane waves:
1. Intensity
2. Phase difference
Spacing of fringes
Interference of two spherical waves
More complicated geometry
),,(cos2²²),,( 2121 zyxAAAAzyxI D
DD rkkzyxzyxzyx
)(),,(),,(),,( 1212
Ref.: B. Dörband
2sin2 e
ns
41
Two Beam Interference
Interference of two plane waves under different directions
Fringe distance s 1212
2
eenkks
42
Interferometry
Basic idea:
- separation of a wave into two beams (test and reference arm)
-every beam surpasses different paths
- superposition and interference of both beams
- analysis of the pattern
Different setups for:
- the beam splitting
- the superposition
- the referencing
Different path lengths
Difference equivalent of one fringe
Measurement of plates:
Haidinger fringes of equal inclination
Newton fringes of equal thickness
Ref: W. Osten
1 1 2 2 wn t n t N tD
2wt
n
43
Classification of Interferometers
Division of amplitude: - Michelson interferometer
- Mach-Zehnder interferometer
- Sagnac interferometer
- Nomarski interferometer
- Talbot interferometer
- Point diffraction interferometer
Division of wavefront: - Young interferometer
- Rayleigh interferometer
Division of source: - Lloyds mirror
- Fresnel biprism
Ref: R. Kowarschik
44
Localization of Fringes
Interference volume for a plate
Interference volume for a wedge
Ref: R. Kowarschik
volume of
interference
fringes
incident
light back side
reflectedfront side
reflected
volume of
interference
fringes
incident
light
back side
reflected
front side
reflected
test surface
beamsplitter
reference surface
here: flat
illumination
to detector
path difference
mRrm
Test by Newton Fringes
Reference surface and test surface with nearly the same radii
Interference in the air gap
Reference flat or curved possible
Corresponds to Fizeau setup
with contact
Broad application in simple
optical shop test
Radii of fringes
45
Ref: W. Osten
Testing with Twyman-Green Interferometer
Short common path,
sensible setup
Two different operation
modes for reflection or
transmission
Always factor of 2 between
detected wave and
component under test
detector
objective
lens
beam
splitter 1. mode:
lens tested in transmission
auxiliary mirror for auto-
collimation
2. mode:
surface tested in reflection
auxiliary lens to generate
convergent beam
reference mirror
collimated
laser beam
stop
46
Testing with Fizeau Interferometer
Long common path, quite insensitive setup
Autocollimating Fizeau surface quite near to test surface, short cavity length
Imaging of test surface on detector
Straylight stop to bloc unwanted light
Curved test surface: auxiliary objective lens (aplanatic, double path)
Highest accuracy
detector
beam
splitter
collimatorconvex
surface
under test
light
source
Fizeau
surface
auxiliary lens
stop
47
Interferograms of Primary Aberrations
Spherical aberration 1
-1 -0.5 0 +0.5 +1
Defocussing in
Astigmatism 1
Coma 1
48
Intensity of fringes
I(x,y,t) intensity of fringes
V(x,y) contrast of pattern
W(x,y) phase function to be found
(x,y,t) reference phase
Rs(x,y) multiplicative speckle noise
IR(x,y,t) additive noise
Tracing of fringes:
- time consuming method, interpolation, indexing of fringes, missing lines
Fourier method:
-wavelet method
- FFT Method
- gradient method
- fit of modal functions
Evaluation of Fringes
),,(),(),,(),(cos),(1),(),,( 0 tyxIyxRtyxyxWyxVyxItyxI RS
49
Real Measured Interferogram
Problems in real world measurement:
Edge effects
Definition of boundary
Perturbation by coherent
stray light
Local surface error are not
well described by Zernike
expansion
Convolution with motion blur
Ref: B. Dörband
50
Interferogram - Definition of Boundary
Critical definition of the interferogram boundary and the Zernike normalization
radius in reality
51
52
Interferometry
General description of the measurement quantity:
superpostion of spatially modulated signal and noise
Io: basic intensity, source
T: transmission of the system, including speckle
: phase, to be found
IN: noise, sensor, electronics, digitization
Signal processing, SNR improvement:
- filtering
- background subtraction
Ref: W. Osten
0( , ) ( , ) ( , ) cos ( , ) ( , )NI x y I x y T x y x y I x y
original signal
filtered signal
background
processed signal
53
Interferometry
perfect interferogram
reduced contrast due
to background intensity
with speckle
with noise
Ref: W. Osten