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Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.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) (Pockelseffect EOlinear En
jk
kjijki EEP )0()()0(2)( )2(
#
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
9.1 Electrooptic Effect
Index Ellipsoid
j
jiji ED Displacement current :
Energy density : ij
jiij EEU 2
1
2
1ED
: for the principal axes
zz
z
yy
y
xx
x DDDU
222
2
1
Put, zyx DU
zDU
yDU
x2/12/12/1
2
1,
2
1,
2
1
12
2
2
2
2
2
zyx n
z
n
y
n
x: Index ellipsoid
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
General expression of index ellipsoid
,j
jiji DE where, : impermeability tensor elementijij )( 1
Energy density : ij
jiij DDU 2
1
2
1ED
Put, zyx DU
zDU
yDU
x2/12/12/1
2
1,
2
1,
2
1
2/12/1
2/12/1
2/12/1
)(2
1,)(
2
1,)(
2
1xzzyyx DD
UzxDD
UyzDD
Uxy
1222 1323122
332
222
11 xzyzxyzyx
3233
2222
1211
1,
1,
1
nnn
21126
231135
232234
2
1,
1,
1
nnn
Put,
11
21
21
2111
62
52
42
2
32
2
22
2
12
xy
nxz
nyz
nz
ny
nx
n
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
Impermeability :
lkl
kijklk
kijkijij EESE )0(
Linear EO Quadratic EO
Kleinman symmetric medium : hkijk
h = 1 2 3 4 5 6
ij = 11 22 33 23,32 13,31 12,21
),,,(1 )3()2()1(3
2zyxjiE
n jjij
i
where, : Electrooptic tensor (element)ij
(9.1-3)
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
3
2
1
636261
535251
434241
333231
232221
131211
62
52
42
32
22
12
1
1
1
1
1
1
E
E
E
n
n
n
n
n
n
(9.1-3)
ex) E=0,
0111
11,
11,
11
062
052
042
2
0322
0222
012
EEE
zEyExE
nnn
nnnnnn
in i
allfor 01
2
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.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
m24
12 632
2
2
2
2
2
xyEn
z
n
y
n
xz
eoo
The Sij matrix is
2
263
632
100
01
01
e
oz
zo
ij
n
nE
En
S
Report : Summary (pp. 333-339)
ijij SS For the principal axis, i
Condition for nontrivial solution (eigenvalue equation) :0
100
01
01
2
263
632
Sn
Sn
E
ESn
e
oz
zo
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
0)(
11 263
2
22
z
oe
ESn
Sn
zo
zoe
En
SEn
Sn
S 6326322
1,
1,
1
1) For S’
011
011
011
3223333232131
2221632323222121
2631221313212111
eo
eoz
zeo
nnSSSS
nnESSSS
Enn
SSSS
arbitrary,0 321
(0,0,1)χ
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
2) For S’’
363232
263163
263163
11
0
0
zoe
zz
zz
Enn
EE
EE
0
0
3
21
)02
1
2
1(
(1,1,0)
,,
χ
3) For S’’’0
0
3
21
)02
1
2
1(
(1,-1,0)
,,
χ
Similarly,
zZyxYyxX ,)(2
1,)(
2
1
111
2
22
6322
632
ez
oz
o n
zyE
nxE
n
New principal axes :
The equation of the index ellipsoid in the new principal coordinate system :
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
9.2 Electrooptic Retardation
For a wave propagating along the z-direction,the equation of the index ellipsoid is
111 2
6322
632
yE
nxE
n zo
zo
263
oz nEAssuming,
zo
oy
zo
ox
En
nn
En
nn
63
3
63
3
2
2
Field components polarized along x’ and y’ propagate as
zEnnctiy
zEnnctiznctix
zoo
zoox
Aee
AeAee
633
633
)2/()/(
)2/()/(])/([
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
Phase difference at the output plane z=lbetween the two components (Retardation) :
,633
c
Vnoyx
lc
nlEV x
xz
, where,
The retardation can also be written as
V
V
V
lEz
6332
where,
on
V
(Halfwave retardation voltage)
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
9.3 Electrooptic Amplitude Modulation
te
te
y
x
cosA
cosA
A)0(
A)0(
y
x
E
E
or, using the complex amplitude notation
222* A2)0()0( yxi EEI EE
A)(
A)(
lE
elE
y
-iΓx
)1(2
A)( -iΓ
oy eE
)]1)(1[(2
A)()(
2* iΓ-iΓ
oyoyo eeEEI
2sinA2 22 Γ
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
The ratio of the output intensity to the input :
V
VΓ
I
I
i
o
2sin
2sin 2
: amplitude modulation
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
Sinusoidal modulation)
tmm sin
2
t
tI
I
mm
mm
i
sinsin12
1
sin24
sin 20
tmm sin12
1
1m
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
9.4 Phase Modulation of Light
Electric field does not change the state of polarization, but merely changes the output phase by
lEc
rnn
c
lzxx 2
6320
'
If the bias field is sinusoidal ; tEE mmz sin
]sincos[ ttAe mout
lErn
c
lErn mm 633063
30
2
where,
: Phase modulation index
)exp()()sinexp( tinJti mn
nm
tni
nnout
meJAe )()(
: side band (harmonics)
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
9.5 Transverse Electrooptic Modulators
Longitudinal mode of modulation : E field is applied along the direction of light propagationTransverse 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
Vr
nnn
c
lezx 63
30
0' 2
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.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
22
1
LC
LRZ
222
22
1
11
11
1
11 so,
CLL
RLCLR
iRZ
Lci
RZ
L
LL
L
Total impedance :
2/12
222
22
2
222
11
1
11
CL
LR
CLL
R
CLL
R
RRZ
L
L
L
Ls
At the resonance [=0=1/(LC)1/2], !LRZ
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
Maximum bandwidth :
CRL
2
1
2
Required power for the peak retardation m :
L
m
R
VP
2
2
where, A : cross-sectional area of the crystal normal to l
,2
)(63
30rn
lEV mmzm
c
RL2
1
263
60
22
4 rnl
AP m
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
Transit-Time Limitation to High-Frequency Electrooptic Modulation
(9.2-4) )/( 6330 crnaaEl
But, if the field E changes appreciably during the transit time through the crystal,
t
t
l
d
dtten
cadzzeat
')'()()(0
Taking e(t) as a sinusoid ; ')'( tim
meEte
Phase change during the transit-time, d=nl/c
where, mmd alEEnca )/(0 : Peak retardation
Reduction Factor, r(Fig. 11-17)
Practically, in order to obtain |r|=0.9,
ti
dm
i
t
t
tim
m
dm
d
m
ei
e
dteEn
cat
1
')(
0
'cnldm / and,2/ d
nlcm 4/)( max
Ex) KDP, n=1.5, l=1cm,
GHz 5)( max m
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.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
)'()'( ttn
ctz
The retardation exercised by this element is given by
dt
t
dttzten
act
')]'(,'[)(
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
)/1(
1)(
)/1(
0mdm
ncciti
ncci
eet
mdm
m
)]')(/('[]'[)'.( ttncktim
zktim
mmmm eEeEzte The traveling modulation field :
where, mmd alEEnca )/(0 : Peak retardation
Reduction factor :
)/1(
)/1( 1mdm
mdm
ncci
nccier
# c/n=cm r = 1
# Maximum modualtion frequency (|r|=0.9) :
)/1(4)( max
mm nccnl
c
case) field (static 4/)( max nlcm
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
9.7 Electrooptic Beam Deflection
Deflection angle inside the crystal (
dx
dn
n
l
Dn
nl
D
y
'
0n )
External deflection angle (By Snell’s law)
dx
dnl
D
nln
'
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
zA Ern
nn 63
30
0 2
zB Ern
nn 63
30
0 2
zErnD
l63
30
Double-prism KDP beam deflector
Number of resolvable spots, N (for a Gaussian beam) :
zbeam
Erln
N 63
30
2
# (9.2-7)63
302
)(lrn
Ez
1422
|63
30
6330
lrn
lrnN VV