Physical Optics...Physical Optics: Content 2 No Date Subject Ref Detailed Content 1 05.04. Wave...
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Physical Optics
Lecture 12: Polarization
2017-05-03
Herbert Gross
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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 G 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
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Introduction
Fresnel formulas
Jones calculus
Further descriptions
Birefringence
Components
Applications
3
Contents
-
Scalar:
Helmholtz equation
Vectorial:
Maxwell equations
Scalar / vectorial Optics
0)(2 rEnko
k
E
H
k
0
Bk
iDk
BEk
jiDHk
EJ
Jk
MHB
PED
r
r
0
0
4
-
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
5
-
Description of electromagnetic fields:
- Maxwell equations
- 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
Basic Notations of Polarization
6
-
Elimination of the time dependence:
Ellipse of the vector E
Different states of polarization:
- sense of rotation
- shape of ellipse
0° 45° 90° 135° 180°
225° 270° 315° 360°
2
2
2
2
2
sincos2
yx
yx
y
y
x
x
AA
EE
A
E
A
E
Polarization Ellipse
7
-
Representation of the state of polarization by an ellipse
Field components
Axes of ellipse: a, b
Rotation angle of the field
Angle of eccentricity
a
btan
Polarization Ellipse
)sinsincos(cos xxxx AE
)sinsincos(cos yyyy AE
cos2
2tan22
yx
yx
AA
AA
x
y
x'
y'
a
b
Ax
Ay
Ex
Ey
8
-
Circular Polarization
Spiral curve of field vector
Superposition of left and right handed circular
polarized light:
resulting linear polarization
Ref: Manset
tx
y
E
Er
El
t
t1
t2
t3
9
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Rotation of plane of polarization with t / z
Phase angle 90°
Generation by l/4 plate out of linear polarized light
Erzeugung :
Polarisator und l / 4 -
Platte
b1
x
y
E
Er
El
t1
El
Er E
t2
b2
SA z
y
x
TA
45°
l / 4 - plate
linear
polarizer
LA
Circular polarized Light
10
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Fresnel Formulas
n n'
incidence
reflection transmission
interface
E
B
i
i
E
B
r
r
E
Bt
t
normal to the interface
i
i'i
a) s-polarization
n n'
incidence
reflection
transmission
interface
B
E
i
i
B
E
r
r
B
E t
t
normal to the interface
i'i
i
b) p-polarization
Schematical illustration of the ray refraction ( reflection at an interface
The cases of s- and p-polarization must be distinguished
11
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Fresnel Formulas
Electrical transverse polarization
TE, s- or -polarization, E perpendicular to incidence plane
Magnetical transverse polarization
TM, p- or p-polarization, E in incidence plane
Boundary condition of Maxwell equations
at a dielectric interface:
continuous tangential component of E-field
Amplitude coefficients for
reflected field
transmitted field
Reflectivity and transmission
of light power
TEe
rTE
E
Er
1 TETEe
tTE r
E
Et 1
' TMTM r
n
nt
nn EE 2211
tt EE 21
TMe
rTM
E
Er
2r
e
PR r
P
2' cos '
cos
t
e
P n iT t
P n i
||
12
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Fresnel Formulas
Coefficients of amplitude for reflected rays, s and p
Coefficients of amplitude for transmitted rays, s and p
tzez
tzezE
kk
kk
inin
inin
innin
innin
ii
iir
'cos'cos
'cos'cos
sin'cos
sin'cos
)'sin(
)'sin(
222
222
tzez
tzezE
knkn
knkn
inin
inin
innnin
innnin
ii
iir
22
22
2222
2222
||'
'
'coscos'
'coscos'
sin'cos'
sin'cos'
)'tan(
)'tan(
tzez
ezE
kk
k
inin
in
innin
in
inin
int
2
'cos'cos
cos2
sin'cos
cos2
'cos'cos
cos2
222
tzez
ezE
knkn
kn
inin
in
innnin
inn
inin
int
22
2
2222||
'
'2
'coscos'
cos2
sin'cos'
cos'2
'coscos'
cos2
13
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Fresnel Formulas
Typical behavior of the Fresnel amplitude coefficients as a function of the incidence angle
for a fixed combination of refractive indices
i = 0
Transmission independent
on polarization
Reflected p-rays without
phase jump
Reflected s-rays with
phase jump of p
(corresponds to r
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Fresnel Formulas: Energy vs. Intensity
Fresnel formulas, different representations:
1. Amplitude coefficients, with sign
2. Intensity cefficients: no additivity due to area projection
3. Power coefficients: additivity due to energy preservation
r , t
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
i
t
r
r
t
R , T
i0 10 20 30 40 50 60 70 80 90
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
T(In)
T(In)
R(In)
R(In)
R , T
i0 10 20 30 40 50 60 70 80 90
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
T
T
R
R
15
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Decomposition of the field strength E
into two components in x/y or s/p
Relative phase angle between
components
Polarization ellipse
Linear polarized light
Circular polarized light
y
x
i
y
i
x
y
x
eA
eA
E
EE
0
xy
0
10E
1
00E
sin
cos0E
iErz
1
2
1
iElz
1
2
1
Jones Vector
222
sincos2
yx
yx
y
y
x
x
AA
EE
A
E
A
E
16
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Jones representation of full polarized field:
decomposition into 2 components
Cascading of system components:
Product of matrices
In principle 3 types of components influencing polarization:
1. Change of amplitude polarizer, analyzer
2. Change of phase retarder
3. Rotation of field components rotator
Jones Calculus
1121 EJJJJE nnn
p
s
trans t
tJ
10
01
cossin
sincos)(rotJ
2
2
0
0
i
i
ret
e
eJ
os
op
ppps
spss
s
p
oE
E
JJ
JJ
E
EEJE ,
System 1 :
J1
E1 System 2 :
J2
E2 System 3 :
J3
E3 System n :
Jn
En-1 EnE4
17
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Rotated component
Rotation matrix
Intensity
Three types of components to change the polarization:
1. amplitude: polarizer / analyzer
2. phase: retarder
3. orientation: rotator
Propagation:
1. free space
2. dielectric interface
3. mirror
)()0()()( DJDJ
cossin
sincos)(D
1
*
12 EJJEI
Jones Matrices
10
012 OPLiPRO eJ
l
p
p
s
TRA t
tJ
0
0
p
s
REF r
rJ
0
0
10
01rJ SP
18
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Birefringence: index of refraction depends on field orientation
Uniaxial crystal:
ordinary index no perpendicular to crystal axis
extra-ordinary index ne along crystal axis
Difference of indices
Jones matrix
Relative phase angle
0
0
i n z
neo i n z
eJ
e
p
l
p
l
2 2
2 2
sin cos sin cos 1( , )
sin cos 1 cos sin
i i
neoi i
e eJ
e e
2
e o
zn n
p
l
oe nnn
Birefringence: Uniaxial Crystal
19
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Descriptions of Polarization
E
Parameter Properties
1
Polarization ellipse
Ellipticity ,
orientation only complete polarization
2
Complex parameter
Parameter
only complete polarization
3
Jones vectors
Components of E
only complete polarization
4
Stokes vectors
Stokes parameter So ... S4
complete or partial
polarization
5
Poincare sphere
Points on or inside the
Poincare sphere only graphical representation
6
Coherence matrix
2x2 - matrix C
complete or partial
polarization
20
-
Description of polarization from the energetic point of view
- So total intensity
- S1 Difference of intensity in x-y linear
- S2 Difference of intensity linear under 45° / 135°
- S3 Difference of circular components
)0,90()0,0(0 IIS
)0,90()0,0(1 IIS
)0,135()0,45(2 IIS
)90,45()90,45(3 IIS
Stokes Vector
22
yxo EES
22
1 yx EES
cos22 yxEES
sin23 yxEES
21
-
Description of a polarization state with Stokes parameterInterpretation:
Components of the field on the Poincare sphere
Also partial polarization is taken into account
Relation
Unequal sign: partial polarization
Stokes vector 4x1
Propagation:
Müller matrix M
2
3
2
2
2
1
2
0 SSSS
3
2
1
0
S
S
S
S
S
Stokes Vector
S
mmmm
mmmm
mmmm
mmmm
SMS
33323130
23222120
13121110
03020100
'
22
-
Linear horizontal / vertical
LInear 45°
Circular clockwise / counter-clockwise
0
0
1
1
S
0
0
1
1
S
0
1
0
1
S
1
0
0
1
S
1
0
0
1
S
Examples of Stokes Vectors
23
-
Partial polarized light:
degree of polarization
p = 0 : un-polarized
p = 1 : fullypolarized
0 < p < 1 : partial polarized
Determination of Stokes parameter:
Unpolarized light
Fully polarized light
Partial Polarization
ges
pol
I
Ip
pS S S
S
12
2
2
3
2
0
pu SS
pSSpS
000 )1(
0321 SSS
2
3
2
2
2
1
2
0 SSSS
24
-
Every point on a uni sphere describes one state of polarization
In spherical coordinates:
Points on z-axis: circular polarized light
Meridian line: linear polarization
Points inside
partial polarization
Sphere of Poincare
2sin
2cos2cos
2sin2cos1
222
z
y
x
zyxr
y
x
z
right handed
circular polarized
left handed
circular polarized
elliptical
polarized
linear
polarized
2
2
25
-
Fully polarized: point
Unpolarized: full surface
Partial polarized: probability distribution, points inside
y
x
z
P
y
x
z
y
x
z
fully polarized un-polarizedpartial
polarized
Partial Polarization on the Poincare Sphere
26
-
Stokes parameter S1 , S2 , S3 :
Componenst along axis directions
Radius of sphere, length of vector: So
Projection into meridional plane:
angle 2 of polarization ellipse
Projection into meridian plane:
eccentricity angle 2
Poincare Sphere and Stokes Vector
So
S1
S2
S3
y
x
z
P
2
2
27
-
Polarization
Polarization of a donat mode in the focal region:
1. In focal plane 2. In defocussed plane
Ref: F. Wyrowski
28
-
Jones matrix changes Jones vector
Müller matrix changes Stokes vector
Change of coherence matrix
Procedure for real systems:
1. Raytracing
2. Definition of initial polarization
3. Jones vector or coherence matrix local on each ray
4. transport of ray and vector changes at all surfaces
5. 3D-effects of Fresnel equations on the field components
6. Coatings need a special treatment
7. Problems: ray splitting in case of birefringence
JCJC'
SMS
'
EJE
'
Propagation of Polarization
29
-
3D calculus in global coordinates according to Chipman
Ray trace defines local coordinate system
Transform between local and
global coordinates
Polarization Raytrace
inLLinjL
in
inL szxsy
ss
ssx
1,1,,
1
11, ,,
'
'
'',',' 11,1,1,1,1, szxsyxx LLinLLL
zzLzL
yyLyL
xxLxL
out
inzinyinx
zLyLxL
xLyLxL
in
syx
syx
syx
T
sss
yyy
xxx
T
11,1,
11,1,
11,1,
,1
,,,
1,1,1,
1,1,1,
,11
''
''
''
,
plane of
incidence
interface
plane
incoming
ray
outgoing
ray
kj-1 kj
xj
yj
y'j
30
-
Embedded local 2x2 Jones matrix
Matrices of refracting surface
and reflection
Field propagation
Cascading of operator matrices
Transfer properties
1. Physical changes
2. Geometrical bending effects
Polarization Raytrace
1,
1,
1,
,
,
,
1
jz
jy
jx
zzyzxz
zxyyxy
zxyxxx
jz
jy
jx
jjj
E
E
E
ppp
ppp
ppp
E
E
E
EPE
121 .... PPPPP MMtotal
100
00
00
,
100
00
00
s
p
rs
p
t r
r
Jt
t
J
100
0
0
2221
1211
,1 jj
jj
J refr
1
,1,1,11
inrefrout TJTP
1
,1,1,11
inbendout TJTQ
31
-
Change of field strength:
calculation with polarization matrix,
transmission T
Diattenuation
Eigenvalues of Jones matrix
Retardation: phase difference
of complex eigenvalues
To be taken into account:
1. physical retardance due to refractive index: P
2. geometrical retardance due to geometrical ray bending: Q
Retardation matrix
Diattenuation and Retardation
EE
EPPE
E
EPT
T
*
*
2
2
minmax
minmax
TT
TTD
2/12/12/12/12/1 wewwJ
i
ret
21 argarg
totaltotalPQR
1
32
-
Müller matrix visualization
Interpretation not trivial
Retardation and diattenuation map
across the pupil
Polarization Zernike pupil aberration
according to M. Totzeck (complicated)
Polarization Performance Evaluation
-
34
Anisotropic Media
Different types of anisotropic media
Ref: Saleh / Teich
-
Classical geometry of birefringent refraction refers on interface plane
Grandfathers method:
Calculation iterative due to non-linear equations in prism coordinates
History: formulas according to Muchel / Schöppe
Only plane setup considered, crystal axis in plane of incidence
Geometry of Raytrace at an Interface
incidentray
e
o
crystalaxis
wavenormal
'
o
''eo
air crystal
a
s'
s
n
35
-
Incident plane wave
Different refractive indices for polarization direction
Important: relative orientation against crystal axis
Arbitrary incidence: splitting of rays
- ordinary ray: perpendicular to optical axis
- extra ordinary ray: in plane of incidence
Ray Splitting in Case of Birefringence
crystal axis
extraordinary ray
ordinary ray
36
-
Birefringence
Different directions of E and D
Ray splitting not identical to wave splitting
37
k / s
P
DE
H , B energy / wave /
Poynting
ray / phase
-
Relationship between electric field and
displacement vector
Linear relationship of tensor-equation
First term, coefficient :
local direction, birefringence
Second term, coefficient :
gradient of field, third order, optical activity, polarization rotated
Third term, :
forth order, intrinsic birefringence, spatial dispersion of birefringence
General:
Due to the tensor properties of the coefficients, all these effects are anisotropic
The field E and the displacement D are no longer aligned
Anisotropic Media
ED r
0
qml
lqmjlmq
ml
lmjlm
l
ljlj EEED,,,
EEssnEssnD
22
38
-
Vanishing solution determinante of the
wave equation
Value of speed of ligth depending on the ray
direction (phase velocity)
Alternative: axis 1/n
Fresnel or Ray Ellipsoid
022
2
22
2
22
2
z
z
y
y
x
x
cc
k
cc
k
cc
k
1/nx
x
y
z
1/ny
1/nz
Ey
electric
field
constant
energy
density
Ex
E
D
displacement
vector
39
-
Inverse matrix of the dielectric tensor:
index or normal ellipsoid
Gives the refractive index as a function of
the orientation
Also possible: ellipsoid of k-values
Index or Normal Ellipsoid
constn
z
n
y
n
x
zyx
2
2
2
2
2
2
z
y
x
r
100
01
0
001
1iin
x
y
z
ny
nz
nx
022
22
22
22
22
22
z
zz
y
yy
x
xx
nn
nk
nn
nk
nn
nk
Dy
electric
fieldconstant
energy
density
Dx
D
E
displacement
vector
40
-
Special case uniaxial crystal:
ellipsoid rotational symmetry
no ordinary direction valid for two directions
ne extra ordinary valid for only one direction
Two cases:
ne > no : positive ( prolate, cigar )
ne < no : negativ ( oblate, disc )
Arbitrary orientation : intersection points
Index Ellipsoid for Uniaxial Crystals
optical
axis
a) positive birefringence ne > no
o-ray
e-ray
optical
axis
b) negative birefringence ne < no
o-ray
e-ray
k2
k3
kn0k0
n0k0
n() k0
n0k0 nek0
41
-
Incident plane wave
Osculating tangential plane at the ordinary index-sphere:
defines normal to o-ray direction
Osculating tangential plane at the index o/e-index ellipsoid:
normal to e-direction
Wave-Optical Construction of the o/e-direction
crystal
axis
incident ray
o - rayeo-ray
velocity
ellipsoid
wavefronts
42
-
Classical uniaxial media used in polarization components:
1. Quartz, positive birefringent, small difference
2. Calcite, negative birefringent, larger difference
Birefringent Uniaxial Media
oe
e
o
quartz calcite
material sign no neo
Calcite negative 1.6584 1.4864
Quarz, SiO2 positive 1.5443 1.5534
43
-
Optical symmetry axis of crystal
material breaks symmetry
Split of rays depends on the
axis orientation
Split of field into two orthogonal
polarisation components
Energy propagation (ray,
Poynting) in general not
perpendicular to wavefront
Refraction with Birefringence
divergent rays
crystal
axis
parallel rays
with phase difference
crystal
axiscrystal
axis
parallel rays
in phase
o-ray e-ray
44
-
Polarizer with attenuation cs/p
Rotated polarizer
Polarizer in y-direction
p
s
LIN c
cJ
10
01
z
y
x
TA
2
2
sincossin
cossincos)(PJ
10
00)0(PJ
Polarizer
45
-
Polarizer and analyzer with rotation
angle
Law of Malus:
Energy transmission
TA
z
y
x
TA
linear
polarizer y
linear
polarizer
E
E cos
2cos)( oII
I
0 90° 180° 270° 360°
Pair Polarizer-Analyzer
parallel
polarizer
analyzer
perpendicular
46
-
Crossed pair of polarizer - analyzer plates
No sample: transmission zero
Polarized sample between the plates:
- spatial resolved rotation of polarization
- analysis of stress and strain
TA z
y
x
TA
linear
polarizer x
linear
polarizer y
analyzer
Stress Analysis: Polarizer / Analyzer
47
-
Phase difference between field
components
Retarder plate with rotation angle
Special value:
l / 4 - plate generates circular polarized light
1. fast axis y
2. fast axis 45°
2
2
0
0
i
i
RET
e
eJ
z
y
x
SA
LA
ii
ii
Vee
eeJ
22
22
cossin1cossin
1cossinsincos),(
iJ V
0
01)2/,0( p
1
1
2
1)2/,4/(
i
iiJ V pp
Retarder
48
-
Rotate the of plane of polarization
Realization with magnetic field:
Farady effect
Verdet constant V
bb
bb
cossin
sincosROTJ
z
y
x
b
VLB
b
Rotator
49
-
Types of LC media:
- nematic long molecules, oriented in one direction
- smectic long molecules, oriented in one direction in several layers
different types A, B, C
- discotic plate-shapes molecules, oriented in a plane
- cholesteric skrew-shaped molecules
Functional principle:
- interaction creates domaines
- orientation of the domaines by voltage
- anisotropic behavior, birefringence
- strong influence of temperature
Parameter to describe the order / entropy:
Application:
- displays
- spatial light modulator
Liquid Crystals
1cos32
1 2 S
50
-
Orientation of the molecules in an external field (voltage)
LCD Cell
moleculesglass plate
field : E = 0 field : E > 0
51
-
Types of Liquid Crystals
nematic smectic A smectic C
52
-
Types of Liquid Crystals
smectic C
cholesteric
53
-
Transmission changes due to polarization
Black-White switch possible
Reflective or refractive
Liquid Crystal Display
polarizer analyzer90° - TN-LC cell
black
white
polarizer LC cell mirror
54
-
epi
illumination trans
illumination
Microscopic Contrast
Ref: M. Kempe
polarization bright field fluorescence
55
-
Differential Interference Contrast
),(
),()1(),(
yxxEeR
yxxEeRyxE
psf
i
psf
i
psfdic
condenser
object
Wollaston
prism
Wollaston
prism
compensator
objective
polarizer
analyzer
shear
distance
x
adjustment
phase
splitting
ratio
R1-R
Point spread function
Parameter:
1. Shear distance
2. Phase offset
3. Splitting ratio
56
-
original / shifted differential splitting
Differential Interference Contrast
Differential splitting:
Large contrast at phase
gradients
57
-
Differential Interference Contrast
Lateral shift : preferred direction
Visisbility depends on orientation of details
shift 45° x-shift (0°) y-shift (90°)
58
-
59
DIC Phase Imaging
Orientation of prisms and shift size determines the anisotropic image formation
Ref.: M. Kempe