Light
and
Matter
Controlling light with matter
Tim Freegarde
School of Physics & Astronomy
University of Southampton
Optical polarization
• light is a transverse wave:
E perpendicular to k
• for any wavevector, there are two field components
• any wave may be written as a superposition of the two polarizations
2
The Fresnel equation
 2E
    E   0 0 2
t
return to derivation of
electromagnetic wave
equation
E r, t   E a exp  it  k  r 
consider oscillatory waves of
definite polarization aω
  UV   U   V  U  V
apply vector identity twice
and simplify
3
The Fresnel equation
 2E
    E   0 0 2
t
return to derivation of
electromagnetic wave
equation
E r, t   E a exp  it  k  r 
consider oscillatory waves of
definite polarization aω
  UV   U   V  U  V
apply vector identity twice
and simplify
4
The Fresnel equation
 2E
    E   0 0 2
t
E r, t   E a exp  it  k  r 
k  k  a   
2
c
k.a  0
k 2    
  UV   U   V  U  V
• for isotropic media
2
a  0
2
c2
• electromagnetic waves are transverse
• the Poynting vector
S
is parallel to the wavevector
k
5
Characterizing the optical polarization
• wavevector insufficient to define
electromagnetic wave
k
• we must additionally define the
polarization vector

 
 
a  ax , a y

• e.g. linear polarization at angle 


i
 
a   cos  , esin
sin

x
z

y
6
Jones vectors
• normalized polarization vector is
known as the Jones vector

a  cos  , ei sin 

• defines polarization state of any
wave of given  and k
• real field corresponds to superposition
of exponential form and complex
conjugate
k


x
z
y
7
Categories of optical polarization
• linear (plane) polarization
• coefficients differ only by real factor
• circular polarization
• coefficients differ only by factor
i
• elliptical polarization
• all other cases
8
Categories of optical polarization
• linear (plane) polarization
• coefficients differ only by real factor
a   cos , sin  
• circular polarization
• coefficients differ only by factor
a   1,  i
• elliptical polarization
• all other cases
i


a  cos  , ei sin 

9
Polarization notation
• circular polarization
• right- or left-handed rotation when
looking towards source
• traces out right- or left-handed
thread
RCP
plane of
incidence
perpendicular
parallel
• linear (plane) polarization
• parallel or perpendicular to plane of
incidence
• plane of incidence contains
wavevector and normal to surface
10
Categories of optical polarization
• complex electric field given by
 E   , E      a  , a   E

x

y


x
y

exp  it  kz
• real electric field corresponds to superposition with complex conjugate
 E   , E     E  a  cost  kz, a  cost  kz   

x

y


x

y
• for monochromatic fields, Jones vector is constant
11
Polarization of time-varying fields
• complex polychromatic electric field given by



  it
  it 
Ex t , E y t     E ax e ,  E a y e 

 

• beating between frequencies causes field to vary with time
• even stabilized lasers have linewidth in the MHz range
• Jones vector may therefore vary on a microsecond timescale – or faster
12
Stokes parameters
• with polychromatic light, the Jones vector varies
 0  E x2  E y2
• we therefore describe polarization through
averages and correlations:
1  E x2  E y2
 2  2 E x E y cos 
•
•
E x , y are the instantaneous field components

is their relative phase
 3  2 E x E y sin 
STOKES PARAMETERS
13
Stokes parameters
• with polychromatic light, the Jones vector varies
E x E x  E y E y  0  E x2  E y2

• we therefore describe polarization through
Ex Ex 
averages and correlations:

•
•

E y E y 1  E x2  E y2
E x E y  E y E x  2  2 E x E y cos 



 field
i Ecomponents
E

E
E x , y are the instantaneous
x y
y E x  3  2 E x E y sin 

is their relative phase
STOKES PARAMETERS
14
Stokes parameters
2
2
 0  IE

E
y
1 x

E x E x  E y E y
• total intensity, I1
1  2EI 2x2 I1 E y2

E x E x  E y E y
• related to horizontally
polarized component, I2
 2  22IE
x EI1
y cos  
3
E x E y  E y E x
• … component polarized at
+45º to horizontal, I3



 3  22IE

I
E
sin


i
E
E

E
4 x y1
x y
y Ex

• … right circularly polarized
component, I4
15
Unpolarized (randomly polarized) light
• average horizontal component = average vertical component
•
average +45º component = average -45º component
•
average RCP component = average LCP component
… = half total intensity
I1  2I 2  2I 3  2I 4
• orthogonal polarizations are uncorrelated
 0 , 1,  2 ,  3   I11, 0, 0, 0 
E x E x  I1 2
E y E y  I1 2
E x E y  0
E y E x  0
16
Degree of polarization
• for partially polarized light, the quantity
12   22   32
V
 02
represents the degree of polarization, where
0 V 1
unpolarized
(randomly polarized)
completely
polarized
17
Completely polarized light
• constant Jones vector
a   a, b 
• Stokes parameters given by
 0 , 1 ,  2 ,  3   I11, a 2  b 2 , ab  ba, iab  ba 
• when simply defining the polarization state, it is common to drop the intensity factor I1
18
• plot the Stokes vector
1
0
1 ,  2 ,  3 
(a) right circularly polarized
[0, 0, 1]
(b) left circularly polarized
[0, 0,-1]
(c) horizontally polarized
[1, 0, 0]
(d) vertically polarized
[-1, 0, 0]
(e) polarized at +45º
[0, 1, 0]
(f) elliptically polarized
(g) unpolarized
(a)
[δ1,δ2,δ3]/δ0
[0, 0, 0]
3
0
(f)
(d)
(g)
1
0
(c)
(e)  2
0
(b)
left elliptically polarized right elliptically polarized
The Poincaré sphere
19
Polarizers
• many optical elements restrict or modify the polarization state of light
• polarization-dependent transmission/reflection • sheet polarizers (Polaroid)
• Nicol, Wollaston prisms etc
• polarizers, polarizing filters, analyzers
• polarization-dependent refractive index
• four categories of physical phenomena
• waveplates, retarders
• polarization-sensitive absorption (dichroism)
• polarization-sensitive dispersion
(birefringence, optical activity)
• reflection at interfaces
• scattering
20
Polarizers
• each mechanism may discriminate between
either linear or circular polarizations
plane of
incidence
• mechanisms depend upon an asymmetry in
the device or medium
perpendicular
parallel
21
Linear polarization upon reflection
• for normal incidence, no distinction between
horizontal and vertical polarizations
• if wavevector makes angle with interface normal,
s- and p-polarizations affected differently
• we consider here the reflection of p-polarized light;
s-polarized beams may be treated similarly
• we resolve the electric field into components
parallel and normal to the interface
• all magnetic field components are parallel to the
interface
E1 t  k1 z 
E2 t  k2 z 
E3 t  k1 z 
x
y
A
B
z
22
Linear polarization upon reflection
combine
forward and
• for normal
incidence,
no distinction between
reflected waves to give total
horizontal and vertical polarizations
fields for each region
• if wavevector makes angle with interface normal,
E//,1  E//,2
s- and p-polarizations affected differently
• weapply
consider
hereconditions
the reflection of p-polarized light;
continuity
for separate
components
s-polarized
beams
may be treatedH
similarly
//,1  H //,2
• we resolve the electric field into components
D  D, 2
parallel and normal to the interface  ,1
hence derive fractional
• alltransmission
magnetic field
components are parallel to the
and reflection
interface
E1 t  k1 z 
E2 t  k2 z 
E3 t  k1 z 
x
y
A
B
z
23
Fresnel equations
1  cos 1
k1 1 cos 2  k2  2  cos 1
1
fields for eachEregion
E3 k1 1  cos 2  k 2  2  cos 1

E1 k1 1  cos 2  k 2  2  cos 1
combine forward
E and
2
• p-polarization
 total
reflected waves to give
2k1
apply continuity conditions
for separate components
• s-polarization
2k1 1 cos 1
E2

E1 k1 1  cos 1  k 2  2  cos 2

k1
cos 1  k2
1  cos 1  k 2
hence derive fractional
E3
k1 1

transmission and reflection
E1
 2  cos 2
 2  cos 2
E1 t  k1 z 
E2 t  k2 z 
E3 t  k1 z 
x
y
A
B
z
24
Linear dichroism
• conductivity of wire grid depends upon
field polarization
• electric fields perpendicular to the wires
are transmitted
• fields parallel to the wires are absorbed
WIRE GRID POLARIZER
25
Linear dichroism
• crystals may similarly show absorption
which depends upon linear polarization
• absorption also depends upon wavelength
• polarization therefore determines crystal
colour
• pleochroism, dichroism, trichroism
TOURMALINE
26
Circular dichroism
• absorption may also depend upon
circular polarization
• the scarab beetle has polarizationsensitive vision, which it uses for
navigation
• the beetle’s own colour depends
upon the circular polarization
SCARAB BEETLE
LEFT CIRCULAR
RIGHT CIRCULAR
POLARIZED LIGHT POLARIZED LIGHT
27
Polarization in nature
• the European cuttlefish also has
polarization-sensitive vision
CUTTLEFISH (sepia officinalis)
• … and can change its colour and polarization!
MAN’S VIEW
CUTTLEFISH VIEW
(red = horizontal polarization)
28
Birefringence
• asymmetry in crystal structure
causes polarization dependent
refractive index
• opposite polarizations follow
different paths through crystal
• birefringence, double refraction
29
Linear polarizers (analyzers)
• birefringence results in different angles of
refraction and total internal reflection
• many different designs, offering different
geometries and acceptance angles
o-ray
e-ray
38.5º
e-ray
o-ray
s-ray
• a similar function results from
multiple reflection
p-ray
30
Waveplates (retarders)
• at normal incidence, a birefringent
material retards one polarization
relative to the other
• linearly polarized light
becomes elliptically polarized
 
2

0 e  l

WAVEPLATE
31
Compensators
• a variable waveplate uses two wedges
to provide a variable thickness of
birefringent crystal
adjust
• a further crystal, oriented with the fast
and slow axes interchanged, allows the
retardation to be adjusted around zero
variable
• with a single, fixed first section, this is a
‘single order’ (or ‘zero order’) waveplate
for small constant retardation
SOLEIL
COMPENSATOR
fixed
32
Optical activity (circular birefringence)
• optical activity is birefringence for
circular polarizations
• an asymmetry between right and left
allows opposing circular polarizations to
have differing refractive indices
• optical activity rotates the polarization
plane of linearly polarized light
• may be observed in vapours, liquids and
solids
CH3
CH2
CH3
CH3 CH3
H
l-limonene
(orange)
H
CH2
r-limonene
(lemon)
CHIRAL MOLECULES
33
Jones vector calculus
• if the polarization state may be represented
by a Jones vector

a  ax , a y 

• then the action of an optical element
may be described by a matrix
 ax   a11 a12  ax 
 
 

 a  a
a 
a
y
   21 22  y 
JONES MATRIX
 a11 a12 
A

a
a
 21 22 
34
Jones vector calculus
 state may be represented
transmission by
• if thepolarization
Aby

1 a

horizontal polarizer
Jones vector
1 0
0 0 
   
a

a
,
a

x
y 
exp
i

0

x
retardation by
A
 the action of an optical element
• 2then

waveplate
0
exp
i

y
may 
be described by a matrix
11 a12 
 cosA  asin
projection onto
a

A3  

rotated axes
21 a22 


sin

cos




 ax   a11 a12  ax 
 
 

 a  a
a 
a
y
   21 22  y 
JONES MATRIX
35
Müller calculus
• field averages and correlations
following optical element depend
linearly upon parameters describing
incident beam
• Müller matrix elements may be
written in terms of Jones matrix
elements, e.g.


b11  a11
a11  a12
a12
  0  b11
  
 1  b21
     b
 2   31
   b
 3   41
b12 b13 b14    0 
 

b22 b23 b24   1 
b32 b33 b34    2 
  
b42 b43 b44    3 
MŰLLER MATRIX


 a21
a21  a22
a22
36
3
0
• the actions of optical materials can be
represented by geometrical transformations
of the Stokes vector in the Poincaré sphere
• optical activity: rotation about a vertical
axis
1
0
2
2
0
left elliptically polarized right elliptically polarized
Müller calculus
37
3
0
• the actions of optical materials can be
represented by geometrical transformations
of the Stokes vector in the Poincaré sphere
• optical activity: rotation about a vertical
axis
• birefringence: rotation about a horizontal
axis
1
0
2
2
0
left elliptically polarized right elliptically polarized
Müller calculus
38