Chapter 7 Electro-optics
Lecture 1 Linear electro-optic effect
7.1 The electro-optic effect
We have seen that light propagating in an anisotropic medium can be decomposed into its
normal modes, or eigenwaves, which can be determined by the index ellipsoid:
x2 y2 z 2


 1.
nx2 n y2 nz2
1
Here nx2   x /  0  11 /  0 . If we define the impermeability tensor ij  ( )ij /  0 , the
index ellipsoid in a general case will be  ij xi x j  1.
In certain crystals, the application of an external electric field will redistribute the charges
in the molecules, thus results in a change in the size and orientation of the index ellipsoid.
This is called the electro-optic effect.
In the presence of an applied electric field E, the index ellipsoid is changed to
ij (E) xi x j  1

ij (E)  ij (0)  rijk Ek  sijkl Ek El  
Here rijk are the linear (Pockels) electro-optic coefficients, and sijkl are the quadratic
(Kerr) electro-optic coefficients.
1
From thermodynamic considerations the linear electro-optic coefficients rijk and the
quadratic electro-optic coefficients sijkl have the following permutation symmetries:
rijk  rjik
sij kl  s ji kl  sijlk  s jilk
We therefore introduce the contracted indices to replace the paired interchangeable indices:
1  (11), 2  (22), 3  (33), 4  (23)  (32), 5  (13)  (31), 6  (12)  (21).
Examples : r(12) k  r( 21) k  r6 k , r( 22) k  r2 k , s(13)( 22)  s(31)( 22)  s52
The permutation symmetry reduces the number of the independent elements of rijk from 27
to 18, and that of sijkl from 81 to 36.
A linear electro-optic effect is also called the Pockels effect, in which the change in the
refractive index n  E . Pockels effect is a second order nonlinear optical phenomenon.
It only occurs in crystals that do not possess the inversion symmetry.
For a crystal with the inversion symmetry, we can equivalently inverse the coordinates.
ij

In the orriginal coordinate s rijk 

Ek

 'ij
 'ij 
In the inversed coordinate s r 'ijk 

  rijk  0.
E 'k  ( Ek ) 

Inverse symmetry  r 'ijk  rijk ,  'ij  ij


2
7.2 The linear electro-optic effect
In an external electric field E the index ellipsoid is deformed into

ij
(0)  ij xi x j  1, or
1
 1  2 1
 1   2 1
 1   2
 1 
 1 
 1 


x



y







 2   z  2 2  yz  2 2  xz  2 2  xy  1.
 2

 2

 2
2
2
 n 1 
 n  2 
 n 3 
 n 4
 n 5
 n 6
 n y
 nz
 nx
For the linear electro-optic effect, the change in the coefficients is given by
3
 1
 2    rij E j , or
 n i j 1
 r11

 r21
  1 
  r31
  2 

 n 
 r
41
1, 2 , 3, 4 , 5, 6 

r
 51
r
 61
r12
r22
r32
r42
r52
r62
r13 

r23 
 Ex 
r33  
 E y 
r43  
Ez
r53  
r63 
Here rij is the linear electro-optic (or Pockels) coefficient, and r is called the electrooptic tensor. Depending on the symmetry of the crystal, many of the elements of the
electro-optic tensor are 0, and some of them have the same or opposite values.
Table 7.1 in the textbook lists the point groups of crystals. We have 7 crystal systems,
which lead to 32 point groups. Table 7.2 lists the non-vanishing linear electro-optic
coefficients of all the point groups. Table 7.3 gives the value of the linear optic-optic
coefficients of some crystals.
3
From Poon and Kim, Engineering Optics with MATLAB.
4
Example 1:
KDP (KH2PO4) crystal (negative uniaxial, 4 2 m point group).
The E-field is in an arbitrary direction.
0

0 0 0




0


0 0 0
E



  1
  0 0 0  x   0

  2 

 E y   
 n 
 r
0 0    r41E x 
41
1, 2 , 3, 4 , 5, 6 

Ez  


0 r

r
E
0
41
y
41




0 0 r 
r E 
63 

 63 z 
x2 y2 z2


 2 r41E x yz  2 r41E y xz  2r63E z xy  1.
no2 no2 ne2
The index ellipsoid is deformed and the xyz axes are no longer the principle axes.
i) For the KDP crystal, if the E field is in the z direction, we have
x2 y2 z 2

  2r63 E z xy  1.
no2 no2 ne2
The index ellipsoid is tilted in the x-y plane.
5
It seems that we can simplify the problem by choosing a new
principle axis system. We rotate the old coordinate system in
the x-y plane counterclockwise by 45° and construct the new
x′,y′,z′ system, that is
1
x' y', y  1 x' y', z'  z
x
2
2
The index ellipsoid in the new coordinate system is
y'
y
x'
x
1
 2 1
 2 z '2
 2  r63Ez  x'  2  r63Ez  y '  2  1.
ne
 no

 no

x '2 y '2 z '2
n3  1 
If we write it as 2  2  2  1, and considerin g n    2 ,
nx ' n y ' nz '
2 n 
1 3

n

n

no r63E z
o
 x'
2

1

then n y '  no  no3r63E z
2

nz '  ne


6
ii) For the KDP crystal, if the applied field is in the x direction, then
x2 y2 z 2
 2  2  2r41E x yz  1.
2
no no ne
The index ellipsoid is tilted in the y-z plane. We rotate the
old coordinate system in the y-z plane counterclockwise by
angle , and construct the new x′,y′,z′ system. Suppose the
index ellipsoid in the new coordinate system is
z
z'
n z'

2no2 ne2 r41E x
 tan 2   n 2  n 2
o
e

n x '  no

2
2
2
2

 1 1 1
x'
y'
z'


1
1
1
2


 1, then 
  2  2   2  2   2 r41E x  
2
n x2' n 2y ' nz2'

 n y ' 2  no ne
 no ne 



2





1
1
1
1
1
1
2
   
  2  2   2 r41E x  
 nz2' 2  no2 ne2

 no ne 




no5ne2 r412 E x2
n y '  no  2( n 2  n 2 ) ,

o
e
For small E x , this leads to 
5 2 2 2
n '  n  ne no r41E x .
e
 z
2( no2  ne2 )
y

ny' y'
If ax 2  by 2  cxy  Ax'2  By '2 , then
tan 2 
c
ab
( a  b)  ( a  b) 2  c 2
A, B 
2
7
Example 2:
LiNbO3 crystal (negative uniaxial, 3m point group).
Suppose the E-field is in the z direction.
 0

 0
  1 
  0
  2 

 n 
  0
1, 2 , 3, 4 , 5, 6 

 r
 51
 r
 22
 r22
r22
0
r51
0
0
r13 
 r13 

 
r13 
 r13 
0


r33    r33 
 0     E z 
0    0 
Ez
0    0 
 

0
0
 1

 1

 1

 2  r13 E z  x 2   2  r13 E z  y 2   2  r33 E z  z 2  1
 no

 no

 ne

x2 y2 z 2
If we write the new index ellipsoid as 2  2  2  1, then the new coefficien ts are
nx n y nz
1
1 3
 1


r
E

n

n

no r13 E z
13 z
x, y
o
2
 n2
n
2
 x, y
o

 1  1  r E  n  n  1 n3r E
33 z
z
e
e 33 z
2
2

n
n
2
e
 z
8
Lecture 2 Electro-optic modulation
7. 3 Electro-optic modulation
In the example of the KDP crystal, if the external E-field is in the z direction, and the light
is propagating in the z direction, the birefringence is
ny '  nx '  no3r63Ez
Suppose the thickness of the plate is d, and the voltage applied is V=Ezd. The phase
retardation is
2
n y '  nx' d  2 no3r63V .



Since the retardation is proportional to the applied voltage, we can consequently convert
the polarization state of the incident light into a desired polarization by choosing the
appropriate voltage. This is called electro-optic retardation.
For practical use, we write the retardation as   
V

, where V  3
is called the
V
2no r63
half-wave voltage, that is the voltage for obtaining a phase retardation of .
Example:
For KDP at =0.55 mm, no=1.50737, r63=10.6×10-12 m/V, we have V=7.5 kV.
9
Electro-optic amplitude modulation
A typical arrangement of an electro-optic amplitude modulator is shown in the figure. A
KDP crystal is placed between two crossed polarizers. Also in the light path there is a
quarter wave plate that introduces a fixed retardation of /2. The input light is polarized in
the x-direction. A voltage of V  Vm sin mt is applied in the z direction of the crystal.
This is called longitudinal electro-optic modulation since the applied electric field is
parallel to the direction of light propagation. The total phase retardation is


2

Vm

sin mt   m sin mt.
V
2
The transmission of the modulator is
I out

 
 1
 sin 2  sin 2   m sin mt   1  sin m sin mt .
I in
2
4 2
 2
For m  1,
I out 1
 1  m sin mt .
I in 2
Therefore a small sinusoidal voltage will cause a sinusoidal modulation of the transmitted
light intensity.
10
11
12
Transverse electro-optic modulation:
We now consider the case where the applied electric field is perpendicular to the
direction of light propagation, which is called the transverse electro-optic modulation.
As shown in the figure, in a KDP crystal, the input light is polarized in the x′-z plane at
45° to each of the axes. The retardation at the output plane is then
no3r63 V
2 
1 3
 2
ne  no l 
transverse 
(nz  nx ' )l 
 l
 ne  no  no r63 E z l 

 
2


d

Here d is the thickness along which the electric field is applied.
2
For comparisio n, longitudinal 
2no3r63V

Compare to longitudinal electro-optic modulation, the advantages of transverse electrooptic modulation are
1) The retardation can be increased by using a longer crystal, or multiple crystals.
2) The filed electrodes do not interfere with the incident light beam.
13
Yariv, Quantum Electronics, 3rd edition.
14
Phase modulation of light
In the case of KDP, if the external field is in the z direction, and the polarization of the
input light is only in the x' direction, then the applied electric field will change the phase of
the light, instead of its polarization. Suppose the field of the input light is Ein  A cos t ,
the applied electric field is Em sin mt , then the field of the output light is

2 
1 3
 
Eout  A cos t 
 no  no r63 Em sin mt d 
 
2
 

(omit the constant phase)
 3


   no r63 Em d , phase modulation index 



 A[ J 0 ( ) cos t  J1 ( ) cos(  m )t  J1 ( ) cos(  m )t
 A cost   sin mt 
 J 2 ( ) cos(  2m )t  J 2 ( ) cos(  2m )t  ]
We see that lights at side bands are generated
15
16
Reading: Lecture 3 Quadratic electro-optic effect
7. 5 Quadratic electro-optic effect
The quadratic electro-optic effect is a third order nonlinear effect, where the change in the
refractive index is proportional to the square of the applied fields. Unlike the linear electrooptic effect, quadratic electro-optic effect occurs in crystals with any symmetry. Using
contracted indices, the index ellipsoid in the presence of quadratic electro-optic effect is
Table 7.4 lists the non-vanishing quadratic electro-optic coefficients of all the point groups.
Table 7.5 gives the values of the quadratic optic-optic coefficients of some crystals.
17
Example 1:
Kerr effect in an isotropic medium. Let the z axis be along the applied electric field.
 s11

 s12
  1 
  s12
  2 

 n 
 
1, 2 , 3, 4 , 5, 6 





s12
s12
s11
s12
s12
s11
 0 
 
 0 
 E 2 
  
( s11  s12 ) / 2
 0 
 
( s11  s12 ) / 2
 0 
( s11  s12 ) / 2  0 
 1
 1
 1
2 2
2 2
2 2
 2  s12 E  x   2  s12 E  y   2  s11E  z  1.
n

n

n

1 3

2
n

n

n
s
E
2
2
2
o
12

x y
z

2
If we write it as


1
,
then

no2
ne2
n  n  1 n 3 s E 2
e
11

2

1
The birefringe nce is ne  no  n 3 s12  s11 E 2  n 3 s44 E 2  KE 2
2
Here K is called the Kerr constant of the substance.
18
Example 2:
Kerr effect in BaTiO3 (m3m). The applied electric field is ( E , E ,0) / 2 .
 s11

 s12
  1 
  s12
  2 

 n 
 
1, 2 , 3, 4 , 5, 6 





s12
s12
s11
s12
s12
s11
1
 1 1
 1
2
2 2
 2  s11E  s12 E  x   2
2
2
n

n
2
 E / 2 
 2
 E / 2 

 0



s44
 0


s44
0


s44  E 2 
1
1

 1

 s11E 2  s12 E 2  y 2   2  s12 E 2  z 2  2s44 E 2 xy  1.
2
2

n

Rotating by 45 in the xy plane gives
1
1 1
2
2
 2  2  s11  s12 E  s44 E
2
 nx ' n

1 1
1
2
2
 2  2  s11  s12 E  s44 E
2
 ny' n
1
1
 2  2  s12 E 2

 nz ' n
x '2 y '2 z '2
 2  2  1, and
2
nx ' n y ' nz '
19
1
, we have
2
n

1 3 1
2
2


n

n

n
s

s
E

s
E
44
 x'
 2 11 12

2




1 3 1
2
2


n

n

n
s

s
E

s
E
 y'
44
 2 11 12

2




1 3
2
nz '  n  n s12 E
2

For s44 E 2 
20