Lecture 15
Chapter IX Electrooptic Modulation
of Laser Beams
Highlights
Propagation of laser beams in crystals with an applied electric field
Index of refraction is proportional to
Controlling the intensity or phase of laser beams …
Linear EO effect
1. Electrooptic (EO) Effect
2. EO Retardation
3. EO Amplitude and Phase Modulations
4. High-Frequency Modulation Considerations
§9.1 Electrooptic Effect
In common crystals:
Birefringence Phenomenon
Index ellipsoid
z (optics axis)
2
2
(0, 0 , ne)
2
x
y
z
 2  2 1
2
nx n y nz
Two linear polarized modes:
ordinary
extraordinary
x
y
z

s
A
(0, -no, 0)
x
B
o
y
(no, 0, 0)
The principal dielectric axes, along which D and E
are parallel
§9.1 Electrooptic Effect
In crystals with an applied E field:
Electrooptic Effect
For example, linear electrooptic effect is the change in the indices of the
ordinary and extraordinary rays that is caused by and is proportional to
an applied electric field.
However, this effect exists only in crystals that do not posses inversion
symmetry
E
Rotate
crystal
n  sE
E
Rotate
crystal
and E
plates
n  sE
s0
n  sE
n  s (  E )
§9.1 Electrooptic Effect
I. The origin of EO Effect
D   0 E  E 2  E 3  ...
No electric field
0
D
P
D  0E
EP
E
Don’t be confused with
p   0 [  (1) E   ( 2 ) EE   ( 3) EEE  ...]
dD
0
2

   2E  3 E  ...
dE
 ,   n02

n 
 n02  2 ' E  3 ' E 2  ...
0
2
n  n0  aE  bE  ...
2
§9.1 Electrooptic Effect
n  n0  aE  bE  ...
2
Index of refraction
w/o E filed
Linear EO effect
(Pockels Effect)
Secondary EO effect
(Kerr Effect)
EO Effect --- the index of refraction change of media due to the
applied electric field.
II. The Index of Refraction Change
W/O E field
x2 y2 z 2
 2  2 1
2
nx n y nz
With E field
 1 2  1 2  1 2  1
 1
 1
x

y

z

2
yz

2
xz

2
 2
 2
 2
 2
 2
 2  xy  1
 n 1
 n 2
 n 3
 n 4
 n 5
 n 6
§9.1 Electrooptic Effect
E0
1
 1
 2
 2
 n 1 E 0 nx
 1
 2
 n 4
E0
 1
 2
 n 2
E 0
 1
 1
 2
 2
 n 5 E 0  n 6
E 0
 1  1 
 2  2
 n i  n i
 1
  2 
 n i
E 0
1
 1 

 2
2
n
n
 3 E 0
z
1
 2
ny
0
E 0
i  1,2,...,6
j  1,2,3
3
 1
  2    rij E j
 n i j 1
x, y , z
rij
Electrooptic tensor
§9.1 Electrooptic Effect
 1 
 2 
 n 1
 1 
 2 
r11
 n 2
r21
 1 
 2 
 n 3  r31
 1 
r41
 2 
 n 4 r51
 1 
r61
 2 
 n 5
 1 
 2 
 n 6
The form of EO tensor for all crystal
symmetry class Table 9-1
r12
r13
r22
r23
r32
r33
r42
r43
r52
r53
r62
r63
Be able to understand this form
E1
E2
E3
Linear EO coefficients of some
commonly used crystals Table 9-2
Be able to link these values to
table 9-1
§9.1 Electrooptic Effect
III. The EO effect in KH2PO4 (KDP)
Symmetry group:
42 m
x2 y2 z 2
 2  2  2r41E x yz  2r41E y xz  2r63 E z xy  1
2
no no ne
Ex  E y  0
x2 y2 z 2
 2  2  2r63 E z xy  1
2
no no ne
Ez  0
z (optics axis)
0 0 0


0 0 0
0 0 0


 r41 0 0 
0 r

0
41


0 0 r 
63 

x
o
y
y
x

§9.1 Electrooptic Effect
 x   cos 
   
 y   sin 
Coordinate Rotation
 sin   x' 
 
cos   y ' 
z  z'
 1
 2  1
 2
 2  2r63 E z sin  cos   x'  2  2r63 E z sin  cos   y '
 no

 no

z '2
 2  2r63 E z (cos 2   sin 2  ) x' y '  1
ne
require
  45
cos   sin   0
2
2
  45
 1
 2  1
 2 z '2
 2  r63 E z  x'  2  r63 E z  y '  2  1
ne
 no

 no

§9.1 Electrooptic Effect
1
1
 2  r63 E z
2
nx ' no
r63 E z  no2
1
1
 2  r63 E z
2
n y ' no
n03
nx '  no  r63 E z
2
n03
n y '  no  r63 E z
2
nz '  nz  ne
IV. The EO effect in LiNbO3 (LN)
Symmetry group:
3m
 0

 0
 0

 0
 r
 51
 r
 22
 r22
r22
0
r51
0
0
r13 

r13 
r33 

0
0 
0 
§9.1 Electrooptic Effect
 1
 2  1
 2
 2  r22 E y  r13 E z  x   2  r22 E y  r13 E z  y
 no

 no

 1
 2
  2  r33 E z  z  2r51E y yz  2r51E x zx  2r22 E x xy  1
 ne

E x  E y  0, E z  0
r13 E z  no2
r33 E z  ne2
 1
 2
 1
 2
2
 2  r13 E z ( x  y )   2  r33 E z  z  1
 no

 ne

n03
nx '  n y '  no  r13 E z
2
ne3
nz '  ne  r33 Ez
2
§9.2 Electrooptic Retardation
KDP crystal
 1
 2  1
 2 z '2
 2  r63 E z  x'  2  r63 E z  y '  2  1
ne
 no

 no

ex '  Ae
i [ t  ( n x ' / c ) z ]
x
or
ex '  Ae
i{t  ( / c )[ no  ( no3 / 2 ) r63E z ] z }
e y '  Ae
and
i{t  ( / c )[ no  ( no3 / 2 ) r6 3E z ] z }
The phase difference at the output plane z=l between two components is
called the retardation
§9.2 Electrooptic Retardation
Phase difference:
no3r63V
   x'   y' 
c
V  Ez l
§9.2 Electrooptic Retardation
z0
ex '  A cos t
0
ex '  A sin t
  /2
Half wave voltage:
e y '  A cos t
ex '   A cos t
 
V
 
V
e y '  A cos t
e y '  A cos t
V 

2no3 r63
Calculate the ADP crystal’s HWV
r63  8.56  10
12
m/V
no ~ 1.5
 ~ 0.5m
V ~ 9000V
§9.3 Electrooptic Amplitude Modulation
At =, the polarization is rotated 90 degree, and is vertical to the
original one. A polarizer after crystal can be used to control the laser
beam intensity which serves as the basis of the EO amplitude
modulation.
At the incident plane
ex '  A cos t
e y '  A cos t
or
E x ' (0)  A
E y ' ( 0)  A
I i (0)  E  E *
| E x ' (0) |2  | E y ' (0) |2  2 A2
§9.3 Electrooptic Amplitude Modulation
Outside the crystal
E x ' (l )  Ae  i
E y ' (l )  A
After polarizer
A  i
E x ' (l ) cos(3 / 4)  
e
2
A
E y ' (l ) cos( / 4) 
2
A i
( E y )o  
(e  1)
2
2
A
*
 i
i
2
2 
I o  [( E y ) o ( E y ) o ] 
(e  1)(e  1)  2 A sin
2
2
Io
2 
2    V 
 sin
 sin   
Ii
2
 2  V 
§9.3 Electrooptic Amplitude Modulation

   m sin( mt )
2
m   (Vm / V )
Io
m
 1
2 
 sin   sin mt   [1  sin( m sin mt )]
Ii
4 2
 2
m  1
Io 1
 (1  m sin  m t )
Ii 2
§9.3 Electrooptic Amplitude Modulation
An optical communication link using an EO modulator
§9.3 Electrooptic Amplitude Modulation
An schematic configuration of EO Q-switch
§9.4 Phase Modulation of Light
Amplitude modulation: by means of EO effect, phase difference convert to the
polarization change, using polarizer to accomplish the intensity modulation
Phase modulation: by means of EO effect, merely changes the output phase
without any change of the polarization
no3r63
l
 x '   nx '  
Ezl
c
2c
§9.4 Phase Modulation of Light
ein  Aeit
E z  Em sin( mt )
eout
no3

 A exp{i[t  (no  r63 Em sin mt )l ]}
c
2
eout  A exp[ i (t   sin mt )]
 no3 r63 Eml  no3 r63 Eml


2c

Phase modulation index
exp( i sin  mt ) 

J
n  
n
( ) exp( in  m t )

eout  A  J n ( )e i (  nm )t
n  
§9.5 Transverse EO Modulators
Longitudinal mode of modulation: electric field is applied along the light
propagation direction
Transverse mode of modulation: electric field is perpendicular to the light
propagation direction
n03
nx '  no  r63 E z
2
nz '  nz  ne
l 
no3r63 V 
   x '   y '  (no  ne ) 

c 
2 d