Waves, Light & Quanta
Tim Freegarde
Web Gallery of Art; National Gallery, London
Sinusoidal waves

z
• simple harmonic motion
yx, t   r sin t  kz
• circular motion
r, where   t  kz
 r cost  kz, r sin t  kz
2
Sinusoidal waves
y
t  t0
x
yx, t   y0 sin t  kx   
at

t  t0 ,
yx, t0   y0 sin  kx    t0 
 y0 sin  2 ~ x    t0 
 2

 y0 sin  
x    t0 
 

• wavenumber
• spectroscopists’
wavenumber
• wavelength
k
2

1
~
 


3
Sinusoidal waves
y
x  x0
t
yx, t   y0 sin t  kx   
at

x  x0,
yx0 , t   y0 sin t    kx0 
• angular frequency
 y0 sin 2  t    kx0 
• frequency
 2

 y0 sin 
t    kx0 
 

• period

2

1



4
Birefringence
• asymmetry in crystal structure
causes two different refractive
indices
• opposite polarizations follow
different paths through crystal
• birefringence, double refraction
5
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
6
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
7
Malus’ law
• amplitude transmission
 cos 

• intensity transmission
 cos 
2
WIRE GRID POLARIZER
8
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
9
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)
10
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
11
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
12
Categories of optical polarization
• linear (plane) polarization
• non-equal components in phase
• circular polarization
• equal components 90° out of phase
• elliptical polarization
• all other cases
13
Polarizing components
LINEAR
CIRCULAR
POLARIZER
(filter/separator)
Tx  Ty
TL  TR
WAVEPLATE
(retarder)
x   y
L  R
14
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
15
Polarization notation
• circular polarization
• right- or left-handed rotation when
looking towards source
• traces out opposite (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
16
Polarization by scattering
cdoswell.com/tips3.htm
17
Brewster’s angle
r
i
i
r 
• reflected light fully (s-) polarized
cos i  sin r
1
 sin i

tan i  
18
Brewster’s angle
i i
r

www.paddling.net/sameboat/archives/sameboat496.html
r
• reflected light fully (s-) polarized
tan i  
19
Malus’ law
• amplitude transmission
 cos 

• intensity transmission
 cos 
2
WIRE GRID POLARIZER
20
Characterizing the optical polarization
• wavevector insufficient to define
electromagnetic wave
• we must additionally define the
polarization vector

a  ax , a y


• e.g. linear polarization at angle 

i
a   cos, esin
 
sin
k


x
z
y
21
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 
22
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
exp i x x 0y  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
23
Birefringence
• asymmetry in crystal structure
causes two different refractive
indices
• opposite polarizations follow
different paths through crystal
• birefringence, double refraction
24
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
25
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
26
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
27
Electromagnetic waves
• light is a transverse wave:
E perpendicular to k
x
Ex
z
By
Ex
• Faraday
• Ampère

y
B
.
d
S
t 
E 

 B.ds  0   J   0 t .dS
E
 B.ds  0 0  t .dS
 E.ds  
x
y
z
By
28
Dielectrics
• atomic electrons move in
response to electric field
• resulting atomic
dipole radiates field
which adds to original
• Faraday
• Ampère
z

B.dS

t
EE

 
B
.
d
s


J



.dS
0  
0 r .dS

t t 

 E.ds  
29