CHAPTER 6-1---ELECTRO-OPTICS
23/4/19Fundamentals of Photonics 1
Chapter 6Chapter 6 ELECTRO-OPTICSELECTRO-OPTICS
CHAPTER 6-1---ELECTRO-OPTICS
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(1) Electro-optic Effect
• refractive index( anisotropic crystal ) change with electric field
• Phase or Polarization change with refractive index
Electro-optic material
Light
Electric field
CHAPTER 6-1---ELECTRO-OPTICS
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Possible application
controllable focal length. Optical scanning device
U
0
Phase modulator
U
Polarization modulation
U
polarizer
analyzer
Light intensity modulator
CHAPTER 6-1---ELECTRO-OPTICS
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Pockels and Kerr Effects
• The refractive index of an electro-optic medium is a function n(E) of the applied electric field E.
21 2
1( )
2n E n a E a E
3 3 21 1( )
2 2n E n rn E n E
Terms higher than the third can be neglected.
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Pockels Effect
31( )
2n E n rn E the third term is negligible
r : Pockels coefficient or the linear electro-optic coefficient range: 10-12 to 10-10
m/VPockels medium or Pockels cell
n
n(E)
0 E
(a)
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Kerr Effects
3 21( )
2n E n n E the second term is negligible
: Kerr coefficient or the quadratic electro-optic coefficient.
Range: 10-18 to 10-14 m2/V2 (for crystal) 10-22 to 10-19 m2/V2 (for liquid)Kerr medium or a Kerr cell
n
n(E)
0 E
(b)
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• Crystal Optics
jiij The index ellipsoid
3,2,1,
1
ji
xxij
jiij
20
1
ij
ijn
k
n2
n1
x
y
z
nb
na
n3
(2) Electro-Optics of Anisotropic Media
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E = (El, E2, E3)
E
( ) _ , , , , 1, 2,3ij ij ijk k ijkl k lk kl
E r E E E i j k l ��������������
{rijk} : linear electro-optic (Pockels) coefficients.
{ijkl} : quadratic electro-optic (Kerr) coefficients.
Pockels and Kerr Effects
CHAPTER 6-1---ELECTRO-OPTICS
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CHAPTER 6-1---ELECTRO-OPTICS
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index ellipsoid equation
where
ij(0) is a diagonal matrix with elements l/n12, l/n2
2, and l/n32
principal refractive indices n1(E), n2(E), and n3(E).
ij
jiij xxE 1)(
k
kijkijij ErE )0()(
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Trigonal 3m Crystals (LiNbO, LiTaO,…)
n1= n2 = no, n3 = ne
Uniaxial crystal
2 2 213 1 2 33 32 2
0
1 1( )( ) ( ) 1
e
r E x x r E xn n
Assuming : E = (0,0, E),
132 20 0
1 1
( )r E
n E n
332 2
1 1
( )e e
r En E n
E
x
y
zOpticaxis
z
x
y
30 13
1n2
r E
3e 33
1n2r E
no
no
ne
CHAPTER 6-1---ELECTRO-OPTICS
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Tetragonal 42m Crystal (e.g., KDP and ADP)22 231 2
63 1 22 20
2 1e
xx xr Ex x
n n
31 0 0 63
1( )
2n E n n r E
32 0 0 63
1( )
2n E n n r E
3( ) en E n
33
'2
'12
'2
'11
'4
sin4
cos
4sin
4cos
xx
xxx
xxx
1'1
')1(')
1( 2
3226320
21632
0
2
xn
xErn
xErn e
x1
x2
x’1x’2
E
x
y
zOpticaxis
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Electra-Optic Modulators and Switches
Phase Modulators
3
00
rn EL
00
0
0
222
nLLnnL
03
dV
L rn
define
0
V
V
V: half-wave voltage
L
V
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Longitudinal modulator Transverse modulator Traveling-wave transverse modulator
d=L several GHz
LV
V
dV
V
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integrated-optical phase modulator
0
V
Input light
Modulated light
Electrodes
V
0
Cross scction
E
Waveguide
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CHAPTER 6-1---ELECTRO-OPTICS
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Dynamic Wave Retarders
EnrnEn 31111 2
1)(
EnrnEn 32222 2
1)(
3 30 1 2 0 1 2 0 1 1 2 2
1[ ( ) ( )] ( ) ( )
2k n E n E L k n n L k rn r n EL
phase retardation
0
V
V
03 3
1 1 2 2
dV
L rn r n
Polarization light
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• Intensity Modulators: Use of a Phase Modulator in an interferometer
2coscos
2
1
2
1 20
iii IIII
2 0( ) cos ( )2 2
VT V
V
VIi Io
Branch 2
Branch 1
Vn V
A
B
CF(V)
0
0.5
1
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VInput light Ii
Modulated light I0
0
An integrated-optical intensity modulator (or optical switch). A Mach-Zehnder interferometer and an electro-optic phase modulator are implemented using optical waveguides fabricated from a material such as LiNbO3
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• Intensity Modulators: Use of a Retarder Between Crossed Polarizers
2 0( ) sin ( )2 2
VT V
V
VV
20if 20
1( ) sin ( ) (0)
4 2 2 2V
V dT VT V T V
V dV V
linear modulation
Vn
V
B
F(V)
0
0.5
1
t
t
Polarizer
s
Ii
Io
Polarizer
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Scanners
3 31 1
2 2
Vn rn E rn
d
scan resolution N
Beam angular divergence:D
0
3
0
1/
2( / )
rn V dN
D
2
VN
V
Large V
+V
-
D
d
L
-V
D
d
L
VV
2V NV
2V NV
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position switch based on electro-optic phase retardation and double refraction.
Electro-optic polarization retator
Birefringent crystal
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Directional Couplers
power-transfer ratio
)0(/)( 102 PLPT
PI(0)
P2(L0)
Waveguide 1
Waveguide 2
z
PI(0)
PI(z) P2(z
)
0 L0
(a)
L0
V
d
Fibers
P1(0)
P2(L)
CHAPTER 6-1---ELECTRO-OPTICS
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CHAPTER 6-1---ELECTRO-OPTICS
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2 2 2 1/ 201( ) sin { [1 ( ) ] }2 2
LT c
=2n/0 : mismatch of the propagation constants
0 00 3 3
0
33
2
C ddV
L rn rn
V0: switching voltage.
C : coupling coefficient.
Lo = /2C
Ernn 3
2
1
0 3 0L
F
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2 2 2 1/ 2
0
1( ) sin { [1 3( ) ] }2 2
VT c
V
0 00 3 3
0
33
2
C ddV
L rn rn
V0: switching voltage.
C : coupling coefficient.Lo = /2C
00V V
F
1
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Spatial Light Modulators
Electrically addressed spatial light modulator
Photo-addressed spatial light modulator
x
y
Incident light
Modulated light
Transmittance T(x,y)
+
+
+
-
-
Electro-optic material
x
y
Write iamge IW(x,y)
Incident light
Modulated light
Transparent electrodes
Mirror
Photoconductive material
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Pockels readout optical modulator (PROM).
BSO
Transparent electrodes
Dichroic reflector of red light
White light (blue)
Incident read light (red)
Modulated light
Polarizing beamsplitter
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Electro-optics of Liquid Crystal
• Electrical Properties of Nematic Liquid Crystals
AnisotropicUniaxial symmetry Optics axis rotate
|| (n||, ne )
(n, no )E
z
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Liquid crystal cell
1
0
0,
2 tan exp( ),2
c
cc
V V
V VV V
V
Phase modulator
2 2
2 2 2
1 cos sin
( ) e on n n
x
y
z
dx
y
z
E
0 0
max 0 0
2 [ ( ) ] /
2 ( ) /e
n n d
n n d
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Dependence of the tilt angle q on the normalized rms voltage
0 0.5 1 1.5 2 2.5 3 3.5 40
10
20
30
40
50
60
70
80
90
(V-Vc)/Vo
0 0.5 1 1.5 2 2.5 3 3.5 40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(V-Vc)/Vo
max
Dependence of the normalized retardation T/Tmax=[n(q) – n0]/(ne-no) on the normalized rms voltage when n0=1.5, for the values of n=ne-no indicated.△
CHAPTER 6-1---ELECTRO-OPTICS
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Phase modulator
Polarization modulator
Nematic Liquid-Crystal Retarders and Modulators
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Reflective light intensity modulator
=/2 (off state) = 0 (on state)
Liquid- crystal
cell
sy
x
Mirror
Polarizer
Incident light
Reflected light
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Twisted Nematic Liquid-Crystal Modulators
Linear polarization direction rotate with liquid crystal twist direction
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x
y
Bright
sDark
x
y
s
(a)
(b)
PolarizerPolarizer
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Reflective twist nematic liquid crystal modulator, normally
45degree twisted
Liquid- crystal
cell
s
Mirror
Polarizer
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Ferroelectric Liquid Crystals
Faster response (us, nematic: ms)
Surface stable Ferroelectric liquid crystal (SSFLC), only on-off state
smectic-C phase
y
x
z
y
x
z
Smectic layers
90°
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Liquid Crystals spatial light modulator
Liquid-Crystal Displays
passive devices relatively slow optical efficiency is limited because of polarization the angle of view is limited
seven-bar-segment LCD
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Optically Addressed Spatial Light Modulators
Hughes liquid-crystal light valve
Transparent electrodes
White light
Incident read light (red)
Modulated light
Polarizing beamsplitter
Light-locking layer
Dielectric mirror
Photoconductor Liquid crystal
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Photorefractive materials
lightfree charge carriersspace-charge distribution
refractive index distribution
LiNbO3
Electric field
(a)
(b)
(c)
(d)
Conduction band
Valence band x
Fe3+
Fe2+
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Simplified theory of photorefractivity
rate of photoionization( ) ( ) ( )D DG x s N N I x
ND : the number density of donorsND
+: the number density of ionized donorsS : the photoionization cross section.
electrons recombination rate( ) ( )R DR x n x N
n(x): electrons density , R is a constant
In equilibrium,R(x) = G(x),
( )( ) ( )D D R DsI x N N n x N
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( ) ( )D D
R D
N Nsn x I x
N
Electric Field ( ) ( ) 0e B e
dnJ e n x E x k T
dx
e : electron mobilityK: Boltzmann’s constantT : temperature.1( )
( )Bk T dn
E xe n x dx
Refractive Index 31( ) ( )
2n x n rE x
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1( )
( )Bk T dI
E xe I x dx
31 1( )
2 ( )Bk T dI
n x n re I x dx
EXAMPLE
0
2( ) (1 cos )
xI x I m
Incident light
max max
sin(2 / ) sin(2 / )( ) , ( )
1 cos(2 / ) 1 cos(2 / )
x xE x E n x n
m x m x
max
2( ) sin
xn x n
If m is small
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x
Fixed-charge density
x
E(x)
x
△ n(x)
x
Free-carrier density
x
I(x)
+ +-
Nonuniform light
Nonuniform light
Photoionization
Diffusion
Recombination at traps
Electric field
Refractive index grating
Response of a photorefractive material to a sinusoidal spatial light pattern
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Applications of the Photorefractive Effect
Two-wave mixing: dynamic holography
Wave 1 (reference)
Wave 2 (object)
Grating
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