Nonlinear Optics Lab . Hanyang Univ.
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
# Nonlinear polarization :
effect) (Pockels
effect EO
linear
n E
jk
k j
ijk
i
E E
P ( ) 2
(2)( 0 ) ( ) ( 0 )
#
Nonlinear Optics Lab . Hanyang Univ.
9.1 Electrooptic Effect
Index Ellipsoid
j
j ij
i E
D
Displacement current :
Energy density :
ij
j i ij
E E
U
2 1 2
1 D E
: for the principal axes
zz z yy
y xx
x D D
U D
2 2 2
2 1
Put, x y Dz
z U U D
y U D
x
2 / 1 2
/ 1 2
/ 1
2 , 1
2 , 1
2
1
2
1
2 2
2 2
2
z y
x
n
z n
y n
x
: Index ellipsoidNonlinear Optics Lab . Hanyang Univ.
General expression of index ellipsoid
,
j
j ij
i
D
E
where, ij (1)ij : impermeability tensor elementEnergy density :
ij
j i ij
D D
U
2 1 2
1 D E
Put, x y Dz
z U U D
y U D
x
2 / 1 2
/ 1 2
/ 1
2 , 1
2 , 1
2
1
2 / 1 2
/ 1 2
/ 1 2
/ 1 2
/ 1 2
/ 1
) 2 (
, 1 ) 2 (
, 1 ) 2 (
1
x z z
y y
x D D
zx U D
U D yz D
U D
xy
1 2
2
2 12 23 13
2 33 2
22 2
11
x
y
z
xy
yz
xz3 33 2
2 22 2
1 11 2
, 1 , 1
1
n n
n
21 12 6 31 2
13 5 32 2
23 4
2
, 1 , 1
1
n n
n
Put,
1 1 1 2
1 2 1 2
1 1
6 2 5
2 4
2 2
3 2 2
2 2 2
1
2
xy
xz n yz n
z n y n
x n n
Nonlinear Optics Lab . Hanyang Univ.
Impermeability :
lkl
k ijkl k
k ijk ij
ij
E S E E
(0)Linear EO Quadratic EO
Kleinman symmetric medium :
ijk
hkh = 1 2 3 4 5 6
ij = 11 22 33 23,32 13,31 12,21
) , , ,
1 3 ( (1) (2) (3)
2 E i j x y z
n i j ij j
where,
ij : Electrooptic tensor (element)(9.1-3)
Nonlinear Optics Lab . Hanyang Univ.
3 2 1
63 62
61
53 52
51
43 42
41
33 32
31
23 22
21
13 12
11
6 2
5 2
4 2
3 2
2 2
1 2
1 1 1 1 1 1
E E E
n n n n n n
(9.1-3)
ex) E=0,
1 0 1
1
1 , 1
1 , 1
1 1
6 0 2 5 0
2 4 0
2
2 3 0
2 2
2 0 2 2
1 0 2
E E
E
E z E y
E x
n n
n
n n
n n
n n
n1 i 0 for all i
2
Nonlinear Optics Lab . Hanyang Univ.
Nonlinear Optics Lab . Hanyang Univ.
Example) EO effect in KH2PO4 (KDP ; negative uniaxial crystal, symmetry group)
When the electric filed is applied along the z-axis, the equation of the index ellipsoid is given by m
2 4
1 2 63
2 2 2
2 2
2 E xy
n z n
y n
x
z e
o o
The Sij matrix is
2 63 2
2 63
0 1 0
1 0 1 0
e o
z
z o
ij
n E n
n E
S
Report : Summary (pp. 333-339)
i j
ij
S
S
For the principal axis,
iCondition for nontrivial solution (eigenvalue equation) :
0 0 1
0
1 0 1 0
2 63 2
2 63
n S n S
E
E n S
e o
z
z o
Nonlinear Optics Lab . Hanyang Univ.
0 ) 1 (
1 2
63 2
2
2
z
o e
E n S
n S
z o
z o
e
n E S n E
n S
S 2 2 63 12 63
1 ,
1 ,
1) For S’
1 0 1
1 0 1 1 0 1
2 3 3 2
3 33 2
32 1 31
2 2 1 2
63 2
3 23 2
22 1
21
2 63 2 1
1 2 3
13 2
12 1 11
e o
e o z
z e
o
n S n
S S
S
n E n
S S
S S
n E S n
S S
S
arbitrary ,
0 3
2
1
(0,0,1)
χ
Nonlinear Optics Lab . Hanyang Univ.
2) For S’’
3 2 63
2 3
2 63 1 63
2 63 1 63
1 1
0 0
z o
e
z z
z z
n E n
E E
E E
0 0
3 2 1
) 2 0 1 2 ( 1 (1,1,0)
,
,
χ
3) For S’’’
0 0
3 2 1
) 2 0 1 2 ( 1
(1,-1,0) , ,
χ Similarly,
z Z y
x Y
y x
X ( ) ,
2 , 1
) (
2 1
1 1 1
2 2 2
2 63 2
2 63
e z
o z
o n
y z n E
x
n
E
New principal axes :
The equation of the index ellipsoid in the new principal coordinate system :
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
1 2
2 63 2
2 63
E y
x n
no
Ez o
z2 63
oz
n
E
Assuming,z o
o y
z o
o x
n E n n
n E n n
63 3
63 3
2 2
Field components polarized along x’ and y’ propagate as
t c n n E z
i y
z E n
n c t
i z
n c t
i x
z o
o
z o
o x
Ae e
Ae Ae
e
6 3 3
6 3 3
) 2 / ( ) / (
) 2 / ( ) / ( ]
) / ( [
Nonlinear Optics Lab . Hanyang Univ.
Phase difference at the output plane z=l
between the two components (Retardation) :
63 ,
3
c V n
oy x
c l l n
E
V
z ,
x
xwhere,
The retardation can also be written as
VV V
l Ez
63
2 3
where,
n
oV
(Halfwave retardation voltage)
Nonlinear Optics Lab . Hanyang Univ.
9.3 Electrooptic Amplitude Modulation
t e
t e
y x
cos A
cos A
A ) 0 (
A ) 0 (
y x
E E
or, using the complex amplitude notation
2 2
* 2
A 2 ) 0 ( )
0
(
x y
i E E
I E E
A ) (
A ) (
l E
e l
E
y
-iΓ
x ( 1)
2 ) A
(Ey o e-iΓ
)]
1 )(
1 2 [(
) A ( ) (
2
*
y o y o -iΓ iΓ
o E E e e
I
sin 2 A
2 2 2Γ
Nonlinear Optics Lab . Hanyang Univ.
The ratio of the output intensity to the input :
V Γ V
I I
i o
sin 2 sin2 2
: amplitude modulation
Nonlinear Optics Lab . Hanyang Univ.
Sinusoidal modulation)
mt
m
sin
2
t
I t I
m m
m m
i
sin sin
2 1 1
2 sin sin2 4
0
1
msin
mt
2
1
1
m
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
l c E
r n n
c l
z x
x 2
63 2 0 '
If the bias field is sinusoidal ; Ez Em sin
mt] sin
cos[ t t
A
e
out
m
n r E l
c l E r
n03 63 m 03 63 m 2
where,
: Phase modulation index )
exp(
) ( )
sin
exp(i t J in mt
n n
m
t n i n
n out
e
mJ A
e
( )
( )
: side band (harmonics)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
# 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
d
r V n n
c n l
e z
x 63
3 0 0
'
2
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
LC parallel circuit,
2 2
2
1
LC
R L
Z
2
22 2 2
1 1 1
1 1
1 1 1
so,
L CL R
L CL iR
R Z
c L R i
Z
L L L L
Total impedance :
2 / 2 1
2 2 2
2 2 2
2 2 2
1 1
1
1
1
L CL R
L CL R
L CL R R R
Z
L L
L L s
At the resonance [=0=1/(LC)1/2], Z RL !
Nonlinear Optics Lab . Hanyang Univ.
Maximum bandwidth :
C R
L
2 1 2
Required power for the peak retardation m :
L m
R P V
2
2where,
A
: cross-sectional area of the crystal normal to l2 , )
(
63 3 0
r l n
E
V
m z m m
c
R
L 21
2 63 6 0 2 2
4 l n r P
mA
Nonlinear Optics Lab . Hanyang Univ.
Transit-Time Limitation to High-Frequency Electrooptic Modulation
(9.2-4) aEl (a
n03r63/c)But, if the field E changes appreciably during the transit time through the crystal,
t
t l
d
dt t n e
a c dz z e a t
' ) ' ( )
( )
(
0
Taking e(t) as a sinusoid ; e(t')Emeimt'
Phase change during the transit-time, d=nl/c
where, 0a(c/n)
dEmalEm : Peak retardation Reduction Factor, r (Fig. 11-17)Practically, in order to obtain |r|=0.9,
t i d
m i t
t
t i m
m d
m d
m
i e e
dt e
n E a c t
1
' )
(
0
'
c
d nl
m
/2, and
d /
nl
m) c/4
( max
Ex) KDP, n=1.5, l=1cm,
GHz 5
) ( max
mNonlinear Optics Lab . 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 )
' ( )
'
( t t
n t c
z
The retardation exercised by this element is given by
t
dt
dt t z t n e
t ac
' )]
' ( ,' [ )
(
Nonlinear Optics Lab . Hanyang Univ.
) /
1 ( ) 1
(
) / 1 ( 0
m d
m
nc c i
t i
nc c i
e e t
m d
m
m
)]
' )(
/ ( ' [ ]
'
)
['.
( t z E
me
i mt kmzE
me
i mt km c n t te
The traveling modulation field :
where, 0a(c/n)
dEmalEm : Peak retardationReduction factor :
) /
1 (
) /
1
(
1
m d
m
m d
m
nc c i
nc c
e
ir
# c/n=cm r = 1# Maximum modualtion frequency (|r|=0.9) :
) / 1 ( ) 4
( max
m
m nl c nc
c
case) field
(static
4 / )
( m maxc nl
Nonlinear Optics Lab . Hanyang Univ.
9.7 Electrooptic Beam Deflection
Deflection angle inside the crystal ( dx dn n
l Dn
n l D
y
'
0
n )
External deflection angle (By Snell’s law)
dx l dn D
l n
n
'
Δn) c(n
TA l
: A ray upper of
me Transit ti
c n TB l : B ray lower of
me Transit ti
n l n T
n T c
B A
y ( )
: B respect to A with
ray of Lag
Nonlinear Optics Lab . Hanyang Univ.
z
A n r E
n
n 63
3 0 0 2
z
B n r E
n
n 63
3 0 0 2
n r E
zD l
63 3
0
Double-prism KDP beam deflector
Number of resolvable spots, N (for a Gaussian beam) :
z beam
E l r
N n
633 0
2
# (9.2-7)
63 3
2 0
)
(
n lr
E
z 1
4 2
| 2
63 3 0 63 3
0