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Birefringence
Halite (cubic sodium
chloride crystal,
optically isotropic)
Calcite
(optically
anisotropic)
Calcite crystal with two polarizers
at right angle to one another
Birefringence was first observed in the 17th century when sailors visiting Iceland brought back to Europe
calcite cristals that showed double images of objects that were viewed through them.
This effect was explained by Christiaan Huygens (1629 - 1695, Dutch physicist), as double refraction of
what he called an ordinary and an extraordinary wave.
With the help of a polarizer we can easily see what these ordinary and extraordinary beams are.
Obviously these beams have orthogonal polarization, with one polarization (ordinary beam) passing
undeflected throught the crystal and the other (extraordinary beam) being twice refracted.
Birefringence
n2  1    
[2]
and
D  E
[3]
as n depends on the direction,  is a tensor
optically isotrop crystal
(cubic symmetry)
uniaxial crystal
(e.g. quartz, calcite, MgF2)
n x  n y  nz
constant phase delay
n x  n y  nz
Birefringence
extraordinary / optic axis
linear anisotropic media:
Di    ij E j
inverting [4] yields:
[4]
j
defining
 ij   ji
principal axes coordinate system:
off-diagonal elements vanish,
D is parallel to E
Dx  11E x
D y   22 E y
E   1D
Dz   33Ez

1

in the pricipal coordinate system  is
diagonal with principal values:
1
i

1
ni2
[5]
Birefringence
the index ellipsoid
a useful geometric representation is:
the index ellipsoid:
 x x
ij i
j
1
[6]
ij
is in the principal coordinate system:
x12 x22 x32
 
1
n12 n22 n32
[7]
uniaxial crystals (n1=n2n3):
cos2   sin 2  


n 2  
n02
ne2
1
na  n0
[8]
nb  n 
n0  n0
n90  ne
Birefringence
double refraction
refraction of a wave has to fulfill the
phase-matching condition
(modified Snell's Law):
nair  sin 1   n   sin  
two solutions do this:
• ordinary wave:
n1  sin 1   n0  sin 0 
• extraordinary wave:
n1  sin 1   n e   sin  e 
Birefringence
uniaxial crystals and waveplates
How to build a waveplate:
input light with polarizations along extraordinary and ordinary axis,
propagating along the third pricipal axis of the crystal
and
choose thickness of crystal according to wavelenght of light
Phase delay difference:  
2

ne  no L
Electro-Optic
Effect
for certain materials n is a function of E,
as the variation is only slightly we can Taylor-expand n(E):
1
n E   n  a1E  a2 E 2  ...
2
Friedrich Carl Alwin
Pockels (1865 - 1913)
Ph.D. from Goettingen
University in 1888
1900 - 1913 Prof. of
theoretical physics in
Heidelberg
linear electro-optic effect
(Pockels effect, 1893):
n E   n 
1
r  n3E
2
r  2
a1
n3
quadratic electro-optic effect
(Kerr effect, 1875):
1


n E  n  s  n3E 2
2
s
a2
n3
Kerr vs Pockels
the electric impermeability (E):

0 1
 2
 n
1
 d 
2  1
3
3 2
2
 E   
  n   3     r  n E  s  n E   r  E  s  E
2
 dn 
n   2

...explains the choice of r and s.
Kerr effect:
Pockels effect:
typical values for s: 10-18 to 10-14 m2/V2
typical values for r: 10-12 to 10-10 m/V
n for E=106 V/m : 10-6 to 10-2 (crystals)
10-10 to 10-7 (liquids)
n for E=106 V/m : 10-6 to 10-4 (crystals)
Electro-Optic Effect
theory galore
from simple picture
 E    0  r  E  s  E 2
[9]
to serious theory:
ij E   ij 0   rijk  Ek   sijkl  Ek El i, j, k , l ,  1,2,3
k
diagonal matrix with
elements 1/ni2
kl
rijk 
ij
Ek
E 0
sijkl
2
1  ij

2 Ek El
E 0
Symmetry arguments ( ij=  ji and invariance to order of differentiation) reduce the
number of independet electro-optic coefficents to:
6x3 for rijk
6x6 for sijkl
a renaming scheme allows to reduce the number of indices to two
(see Saleh, Teich "Fundamentals of Photonics")
and crystal symmetry further reduces the number of independent elements.
[10]
Pockels Effect
doing the math
How to find the new refractive indices:
• Find the principal axes and principal refractive
indices for E=0
• Find the rijk from the crystal structure
• Determine the impermeability tensor using:
ij E   ij 0   rijk Ek
k
• Write the equation for the modified index ellipsoid:
 ( E ) x x
ij
i
j
1
ij
• Determine the principal axes of the new index
ellipsoid by diagonalizing the matrix ij(E) and find
the corresponding refractive indices ni(E)
• Given the direction of light propagation, find the
normal modes and their associated refractive
indices by using the index ellipsoid (as we have
done before)
Pockels Effect
what it does to light
Phase retardiation (E) of light after passing through a Pockels Cell of
lenght L:
 E  
with
this is
with
n E   n 
V
d

na E   nb E L
[11]
1
r  n3E
2
 E  
E
2
2

[12]
na  nb L 
1 2
ra na3  rb nb3 EL [13]
2 


a Voltage applied between two surfaces of the crystal
the retardiation is finally:
V
  0  
V
0 
2

na  nb L
d

V 
L ra na3  rbnb3
[14]
Pockels Cells
building a pockels cell
Construction
Longitudinal Pockels Cell (d=L)
•
V 

r  n3
• V scales linearly with 
• large apertures possible
Transverse Pockels Cell
•
V 
d 
L r  n3
• V scales linearly with 
• aperture size restricted
from Linos Coorp.
Pockels Cells
Dynamic Wave Retarders / Phase Modulation
Pockels Cell can be used as
dynamic wave retarders
Input light is vertical, linear polarized
with rising electric field (applied Voltage) the
transmitted light goes through
• elliptical polarization
• circular polarization @ V/2 (U  /2)
• elliptical polarization (90°)
• linear polarization
(90°)
@ V
  0  
V
V
Pockels Cells
Phase Modulation
Phase modulation leads to
frequency modulation
definition of frequency:
2  f t  
d t 

dt
[15]
with a phase modulation
2  f t  
d t 
d t 
 
dt
dt
 t   m sin t 
 frequency modulation at frequency 
with 90° phase lag and peak to peak
excursion of 2m
 Fourier components:
power exists only at discrete optical
frequencies k 
Pockels Cells
Amplitude Modulation
• Polarizer
guarantees, that
incident beam is
polarizd at 45° to the
pricipal axes
• Electro-Optic Crystal
acts as a variable
waveplate
• Analyser
transmits only the
component that has
been rotated
-> sin2 transmittance
characteristic
Pockels Cells
the specs
• Half-wave Voltage
O(100 V) for transversal cells
O(1 kV) for longitudinal cells
• Extinction ratio
up to 1:1000
• Transmission
90 to 98 %
• Capacity
O(100 pF)
• switching times
O(1 µs)
(can be as low as 15ns)
preferred crystals:
• LiNbO3
• LiTaO3
• KDP (KH2PO4)
• KD*P (KD2PO4)
• ADP (NH4H2PO4)
• BBO (Beta-BaB2O4)
longitudinal cells
Pockels Cells
temperature "stabilization"
an attempt to compensate thermal birefringence
Electro Optic
Devices
Liquid Crystals
Faraday Effect
Optical activity
Faraday Effect
Photorefractive Materials
Acousto Optic