Download Creation of Colloidal Periodic Structure

Chapter 9. Electrooptic Modulation of Laser Beams
9.0 Introduction
Electrooptic effect :
# Effect of change in the index of refraction of medium (crystal)
by an external (DC) electric field
# n  E  linear EO effect (Pockels effect)
# Nonlinear polarization :
( 2)
Pi ( )  2  ijk
(    0) E j ( ) Ek (0)
jk
Nonlinear Optics Lab.
Hanyang Univ.
9.1 Electrooptic Effect
Index Ellipsoid
Displacement current : Di    ij E j
j
Energy density : U 
1
1
D  E    ij Ei E j
2
2 ij

2
1  Dx2 D y Dz2 
U      : for the principal axes
2   xx  yy  zz 
1/ 2
1/ 2
1/ 2
1
1
1
Put, x  Dx , y   Dy , z   Dz
 2U 
 2U 
 2U 

x2 y2 z 2
 2  2 1
2
nx n y nz
: Index ellipsoid
Nonlinear Optics Lab.
Hanyang Univ.
General expression of index ellipsoid
Ei ij D j ,
where, ij  ( 1 )ij : impermeability tensor element
j
Energy density : U 
1
1
D  E  ij Di D j
2
2 ij
1/ 2
Put,
1/ 2
1/ 2
 1 
 1 
 1 
x
 Dx , y 
 Dy , z 
 Dz
 2U 
 2U 
 2U 
1/ 2
1/ 2
1/ 2
 1 
 1 
 1 
1/ 2
1/ 2
1/ 2
xy
 ( Dx Dy ) , yz 
 ( Dy Dz ) , zx 
 ( Dz Dx )
 2U 
 2U 
 2U 
 11x 2 22 y 2 33 z 2 212 xy223 yz 213 xz1
Put, 11  
1 
 1 
 1 
,


,





 2
22
33
2
2
 n 1
 n 2
 n 3
1
1
1
 2   23  32 ,
 2   13  31 ,
 2   12  21
 n 4
 n 5
 n 6
1
1
1
1
1
1
  2  x 2  2  y 2  2  z 2 2 2  yz 2 2  xz2 2  xy1
 n 1
 n 2
 n 3
 n 4
 n 5
 n 6
Nonlinear Optics Lab.
Hanyang Univ.
Impermeability :
 ij   ij(0)    ijk E k   S ijkl Ek El    
k
kl
Linear EO Quadratic EO
Kleinman symmetric medium :
1 3
  2   ij E j
 n i j
where,
 ijk   hk
h=
1
2
3
4
5
6
ij =
11
22
33
23,32
13,31
12,21
(1) ( 2 ) ( 3)
(i, j  x , y , z )
(9.1-3)
 ij : Electrooptic tensor (element)
Nonlinear Optics Lab.
Hanyang Univ.
(9.1-3) 
 11
 1 
 2 
 n 1
 1 
 21
 2 
 n 2
 1 
 31
 2 
 n 3 
 1 
 2 
 41
 n 4
 1 
 2 
 51
 n 5
 1 
 2 
 61
 n 6
1
0 for all i
2 
 n i
ex) E=0, 
 12  13
 22  23
1
 1

,
 2
2
n
n
 1 E0 x
 32  33
 1
 2
 n 4
 42  43
E1
E2
E3
 1
 2
 n 2

E 0
1
,
n y2
 1
 1
 2 
 2 
n 5 E0  n  6
E 0 
1
 1

 2
2
 n 3 E0 nz
0
E 0
 52  53
 62  63
Nonlinear Optics Lab.
Hanyang Univ.
Nonlinear Optics Lab.
Hanyang Univ.
Example) EO effect in KH2PO4 (KDP ; negative uniaxial crystal, 42m symmetry group)
When the electric filed is applied along the z-axis, the equation of the index ellipsoid is given by
x2 y2 z 2
 2  2  2 63 Ez xy  1
2
no no ne
 1
 2
 no
S ij   63 E z


 0

The Sij matrix is
For the principal axis,  i
 63 E z
1
no2
0

0

0

1 
ne2 
Report : Summary (pp. 333-339)
 Sij  j S i
Condition for nontrivial solution (eigenvalue equation) :
1
 2 S
 no
 E
 63 z

 0

Nonlinear Optics Lab.
 63 E z
1
S
no2
0



0 0


1
 S
2
ne

0
Hanyang Univ.
2

 1
 1

2
  2 S  2 S  ( 63 Ez ) 0

 ne
 no 
 S 
1
1
1





,
S



E
,
S

 63 E z
63 z
ne2
no2
no2
1) For S’
S111  S12  2  S13  3  S 1

S 211  S 22  2  S 23  3  S  2

S311  S32  2  S33  3  S  3

 1 1
 2  2  1   63 E z  2  0
 no ne 
 1 1
 63 E z 1   2  2   2  0
 no ne 
 1 1
 2  2   3  0
 no ne 
 1 2 0,  3 arbitrary
χ   (0,0,1)
Nonlinear Optics Lab.
Hanyang Univ.
2) For S’’
  63 E z 1   63 E z  2  0
 63 E z 1   63 E z  2  0
 1

1

 






E
3
63 z   3
 n2
ne2
 o

1   2  0
 3  0
 χ(1,1,0)
1 1
( , ,0)
2 2
Similarly,
3) For S’’’
1  2  0
 3  0
New principal axes :
X 
 χ (1,-1,0)
1
1
( , ,0)
2
2
1
1
(x  y ) , Y 
(x  y ) , Z  z
2
2
The equation of the index ellipsoid in the new principal coordinate system :
1
 2 1
 2 z2
 2   63 Ez  x   2   63 Ez  y  2  1
ne
 no

 no

Nonlinear Optics Lab.
Hanyang Univ.
9.2 Electrooptic Retardation
For a wave propagating along the z-direction,
the equation of the index ellipsoid is
1

1

 2   63 Ez  x2   2   63 Ez  y2  1
 no

 no

2
Assuming,  63 E z  no
Field components polarized along x’ and y’ propagate as
ex  Ae
i[t ( / c ) n x z ]
e y  Ae 

 Ae 

no3
nx no   63 E z
2
no3
ny no   63 E z
2

i t ( / c ) no  ( no3 / 2 ) 63E z z

i t ( / c ) no ( no3 / 2 ) 63E z z
Nonlinear Optics Lab.
Hanyang Univ.
Phase difference at the output plane z=l
between the two components (Retardation) :
 x  y 
where,
no3 63V
c
,
V  E z l ,  x 
nx
c
l
The retardation can also be written as
Ezl
V
 

V
V
where,
V 

2no3 63
(Halfwave retardation voltage)
Nonlinear Optics Lab.
Hanyang Univ.
9.3 Electrooptic Amplitude Modulation
ex  A cos t
e y  A cos t
or, using the complex amplitude notation
E x (0)  A
E y (0)  A
A -iΓ
(e  1)
2
Ex (l )  Ae-iΓ
( E y )o 
E y (l )  A
A 2 -iΓ
I o ( E y ) o ( E )  [(e 1)(eiΓ 1)]
2
Γ
2A 2 sin 2
2
*
y o
2
I i  E  E*  Ex (0)  E y (0)  2A 2
2
Nonlinear Optics Lab.
Hanyang Univ.
The ratio of the output intensity to the input :
   V 
Io
2 Γ
 sin
 sin   
Ii
2
 2  V 
: amplitude modulation
Nonlinear Optics Lab.
Hanyang Univ.
Sinusoidal modulation)

   m sin mt
2
I0
 

 sin 2   m sin mt 
Ii
4 2


1
1  sin m sin mt 
2
1
 1m sin  m t 
2
m 1
Nonlinear Optics Lab.
Hanyang Univ.
9.4 Phase Modulation of Light
Electric field does not change the state of polarization,
but merely changes the output phase by
x '  
l
nx  
c
n02 r63
2c
Ez l
If the bias field is sinusoidal ; E z  Em sin  m t
eout  A cos[t   sin mt ]
 n03r63 Eml  n03r63 Eml
where,  

2c

exp( i sin mt ) 
: Phase modulation index

J
n  
n
( ) exp( inmt )

 eout  A J n ( )e i ( n m )t
: side band (harmonics)
n
Nonlinear Optics Lab.
Hanyang Univ.
9.5 Transverse Electrooptic Modulators
Longitudinal mode of modulation : E field is applied along the direction of light propagation
Transverse mode of modulation : E field is applied normal to the direction of light propagation
l 
n03  V 
  x '  z  n0  ne   r63  
c 
2  d 
# Transverse mode is more desirable :
1) Electrodes do not interfere with the optical beam
2) Retardation (being proportional to the crystal length) can be increased by use of longer crystal
3) Can make the crystal have the function of /4 plate
Nonlinear Optics Lab.
Hanyang Univ.
9.6 High-Frequency Modulation Considerations
If Rs > (oC)-1, most of the modulation voltage drop is across Rs
 wasted !
Solution : LC resonance circuit + Shunting resistance, RL >>Rs
 L 
LC parallel circuit, Z  R 

2
 1 LC 
2
2
1
so, Z1 
1 
1 
i c 
RL 
L 




RL  2CL1
RL i
L
Z1 
2
2
R 
1 L   2CL1
 L 
2
Total impedance :
2
2 1/ 2


2
 

R
2
L

 

 CL  1  
R
 
 

L
L
Z   Rs 
 
 
2
2
2
2
R
R





1   L   2CL  1  1   L   2CL  1  
   L 
 

 L 

At the resonance [=0=1/(LC)1/2], Z  RL !
Nonlinear Optics Lab.
Hanyang Univ.

1

Maximum bandwidth :
2 2RLC
Required power for the peak retardation m :
Vm2
P
2 RL
Vm ( E z ) m l 
m
,
3
2 n0 r63
1
c
2 RL
m2 2 A
 P
4 ln06 r632
where,
A
: cross-sectional area
of the crystal normal to l
Nonlinear Optics Lab.
Hanyang Univ.
Transit-Time Limitation to High-Frequency Electrooptic Modulation
(9.2-4)  aEl (an03 r63 /c)
But, if the field E changes appreciably during the transit time through the crystal,
l
t
c
(t )  a  e( z )dz  a  e(t ' )dt '
n t  d
0
Taking e(t) as a sinusoid ; e(t ') Em ei mt '
Phase change during the transit-time, d=nl/c
t
c
(t )  a Em  eimt ' dt '
n t  d
1  e im d
 0 
 im d
 i m t
e

Practically, in order to obtain |r|=0.9,
Reduction Factor, r
(Fig. 11-17)
 m d  /2, and  d nl /c
 ( m ) max c/4nl
Ex) KDP, n=1.5, l=1cm,
where, 0 a(c/n) d Em alEm : Peak retardation
Nonlinear Optics Lab.
 ( m ) max 5 GHz
Hanyang Univ.
Traveling wave Modulators
: matching the phase velocities of the optical and modulation fields
by applying the modulation signal in the form of a traveling wave
Consider an element of the optical wavefront that enters the crystal at z=0 at time t
c
z (t ' )  (t 't )
n
t 
ac d
The retardation exercised by this element is given by (t )
e[t ',z (t ')]dt '
n
Nonlinear Optics Lab.

t
Hanyang Univ.
The traveling modulation field :
e(t '.z ) Em ei[ mt ' km z ]  Em ei[ mt ' km ( c/n )( t 't )]
 (t )  0e
i mt
 eim d (1c/ncm ) 1 


 i m d (1c/ncm ) 
where, 0 a(c/n) d Em alEm : Peak retardation
Reduction factor :
r
eim d (1c / ncm )  1
im d (1c / ncm )
# c/n=cm  r = 1
# Maximum modualtion frequency (|r|=0.9) :
c
( m ) max 
4nl (1c/ncm )
 ( m ) max c/4nl (static field case)
Nonlinear Optics Lab.
Hanyang Univ.
9.7 Electrooptic Beam Deflection
Deflection angle inside the crystal ( n0 )
'  
y
ln
l dn

 
D
Dn
n dx
External deflection angle (By Snell’s law)
   ' n  l
n
dn
 l
D
dx
Nonlinear Optics Lab.
Hanyang Univ.
Double-prism KDP beam deflector
n03
n A n0  r63 E z
2
n03
nB n0  r63 E z
2
l
   n03 r63 E z
D
Number of resolvable spots, N (for a Gaussian beam) :
n03l
N

r63 E z
 beam 2

# (9.2-7)
E z ( )

2n03lr63
n03lr63 

 N |V V 
 1
3
2 2n0 lr63 4
Nonlinear Optics Lab.
Hanyang Univ.