Download CHAPTER 6-1---ELECTRO

CHAPTER 6-1---ELECTRO-OPTICS
Chapter 6
ELECTRO-OPTICS
Fundamentals of Photonics
2017/5/14
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CHAPTER 6-1---ELECTRO-OPTICS
(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
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CHAPTER 6-1---ELECTRO-OPTICS
Possible application
0

U
Optical scanning device
controllable focal length.
Phase modulator
analyzer
polarizer
U
U
Polarization modulation
Fundamentals of Photonics
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Light intensity modulator
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CHAPTER 6-1---ELECTRO-OPTICS
Pockels and Kerr Effects
• The refractive index of an electro-optic medium is
a function n(E) of the applied electric field E.
1
n( E )  n  a1E  a2 E 2  
2
1 3
1 3 2
n( E )  n  rn E   n E 
2
2
Terms higher than the third can be neglected.
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CHAPTER 6-1---ELECTRO-OPTICS
Pockels Effect
1 3
n( E )  n  rn E the third term is negligible
2
n(E)
r : Pockels coefficient
or the linear electro-optic
coefficient
range: 10-12 to 10-10 m/V
Pockels medium or Pockels cell
n
0
E
(a)
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CHAPTER 6-1---ELECTRO-OPTICS
Kerr Effects
1 3 2
n( E )  n   n E the second term is negligible
2
n(E)
: 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
n
(for liquid)
Kerr medium or a Kerr cell
0
E
(b)
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CHAPTER 6-1---ELECTRO-OPTICS
(2) Electro-Optics of Anisotropic Media
• Crystal Optics

z
n3
x xj 1
k
ij i
ij
i, j  1,2,3
na
n2
1

0
ij    2
nij
y
n1
x
ij   ji
Fundamentals of Photonics
nb
The index ellipsoid
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CHAPTER 6-1---ELECTRO-OPTICS
Pockels and Kerr Effects
E = (El, E2, E3)
ij ( E )  ij   rijk Ek _  ijkl Ek El , i, j, k , l  1, 2,3
k
E
kl
{rijk} : linear electro-optic (Pockels)
coefficients.
{ijkl} : quadratic electro-optic (Kerr)
coefficients.
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CHAPTER 6-1---ELECTRO-OPTICS
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CHAPTER 6-1---ELECTRO-OPTICS
index ellipsoid equation

ij
( E ) xi x j  1
ij
where
ij ( E )  ij (0)   rijk Ek
k
ij(0) is a diagonal matrix with elements l/n12, l/n22, and l/n32
principal refractive indices n1(E), n2(E), and n3(E).
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CHAPTER 6-1---ELECTRO-OPTICS
Trigonal 3m Crystals (LiNbO, LiTaO,…)
z
Uniaxial crystal
Optic
axis
n1= n2 = no, n3 = ne
Assuming : E = (0,0, E),
E
1
1
2
2
( 2  r13 E )( x1  x2 )  ( 2  r33 E ) x32  1
n0
ne
y
x
z
n
1 3
n e r33 E
2
e
1
1
 2  r13 E
2
n0 ( E ) n0
n
1
1

 r33 E
2
2
ne ( E ) ne
y
o
x
n
o
1 3
n 0 r13 E
2
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CHAPTER 6-1---ELECTRO-OPTICS
Tetragonal 42m Crystal (e.g., KDP and ADP)
x12  x22 x32
 2  2r63 Ex1 x2  1
2
n0
ne
x1  x1 cos
'
x2  x1 cos
'

4

4
 x2 sin
'
 x2 sin
'
Optic
z axis
x’2

4

x2
x’1
E
y
4
x1
x
x3  x'3
2
1
1
1 2
2
( 2  r63 E ) x1 ' ( 2  r63 E ) x2 '  2 x'3  1
n0
n0
ne
1 3
n1 ( E )  n0  n0 r63 E
2
1 3
n2 ( E )  n0  n0 r63 E
2
n3 ( E )  ne
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CHAPTER 6-1---ELECTRO-OPTICS
Electra-Optic Modulators and Switches
Phase Modulators

2nL
0
  0  

2n0 L
0

2nL
0
V
L
rn3 EL
0
V
d 0
  0  
define V 
3
V
L rn
V:
half-wave voltage
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CHAPTER 6-1---ELECTRO-OPTICS
V
d
V
L
V
V
Longitudinal modulator Transverse modulator
d=L
Fundamentals of Photonics
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Traveling-wave
transverse modulator
several GHz
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CHAPTER 6-1---ELECTRO-OPTICS
Electrodes
Input
light
V
Waveguide
0
V
Modulated
light
E
0
Cross scction
integrated-optical phase modulator
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CHAPTER 6-1---ELECTRO-OPTICS
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CHAPTER 6-1---ELECTRO-OPTICS
Dynamic Wave Retarders
1 3
n1 ( E )  n1  r1n1 E
2
1
3
n2 ( E )  n2  r2 n2 E
2
phase retardation
1
  k0 [n1 ( E )  n2 ( E )]L  k0 (n1  n2 ) L  k0 (r1n13  r2 n23 ) EL
2
V
0
d
  0  
V 
3
3
L
r
n

r
n
V
1 1
2 2
Polarization light
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CHAPTER 6-1---ELECTRO-OPTICS
• Intensity Modulators: Use of a Phase
Modulator in an interferometer
Branch 2
Ii
Branch 1
Io
V
F(V)
1
C
B
0.5
A
0
I0 
V
1
1

I i  I i cos   I i cos 2
2
2
2
 V
2 0
T (V )  cos (
Fundamentals of Photonics
Vn
2

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2 V
)
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CHAPTER 6-1---ELECTRO-OPTICS
Input
light Ii
V
0
Modulated
light I0
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
Fundamentals of Photonics
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CHAPTER 6-1---ELECTRO-OPTICS
• Intensity Modulators: Use of a Retarder
Between Crossed Polarizers
Io
s
F(V)
1
Polarizer
0.5
0
V
t
V
n
Ii
Polarizer
 V
2 0
T (V )  sin ( 
)
2 2 V
if
B
t
0  
2 T (V )  sin 2 (    V )  T (0)  dT
4 2 V
dV
V  V
1  V

V 0V 
2 2 V
linear modulation
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V  2 NV
CHAPTER 6-1---ELECTRO-OPTICS
Scanners
d
-
1
1
V
  n    rn3 E    rn3
2
2
d


D
scan resolution N
Beam angular divergence:
V
+
V
1
3

rn
V /d

N
2

(0 / D)
L
d

-V
  0 D
D

V
V  2 NV
N
2V
Large V
L
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CHAPTER 6-1---ELECTRO-OPTICS
Electro-optic
polarization retator
Birefringent
crystal
position switch based on electro-optic phase
retardation and double refraction.
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CHAPTER 6-1---ELECTRO-OPTICS
Directional Couplers
PI(0)
Waveguide 1
P1(0)
P2(L0)
Waveguide 2
Fibe
rs
V
PI(0)
L
PI(z)
0
d
P2(z)
L0
0
P2(L)
z
(a)
power-transfer ratio
T  P2 ( L0 ) / P1 (0)
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CHAPTER 6-1---ELECTRO-OPTICS
Fundamentals of Photonics
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CHAPTER 6-1---ELECTRO-OPTICS
 L0 2 1/ 2
1
T  ( ) sin c { [1  (
) ] }
2
2


2
2
=2n/0 : mismatch of the propagation constants
1 3
n   rn E
2
d 0
F
3 C 0 d
V0  3

3
L0 2rn
 rn3
V0: switching voltage.
0
3
 L0
C : coupling coefficient.
Lo = /2C
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CHAPTER 6-1---ELECTRO-OPTICS
d 0
3 C 0 d
V0: switching voltage.
V0  3

3
3
L0 2rn
 rn
Lo = /2C C : coupling coefficient.

1
V 2 1/ 2
T  ( ) sin c { [1  3( ) ] }
2
2
V0
2
2
F
1
0
Fundamentals of Photonics
V0
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V
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CHAPTER 6-1---ELECTRO-OPTICS
Spatial Light Modulators
y
Modulated
light
+
Transmittance T(x,y)
Incident
light
-
x
Incident
light
+
Modulated
light
+
Transparent
electrodes
Write iamge
IW(x,y)
Mirror
y
Electro-optic
material
x
Electrically addressed
spatial light modulator
Fundamentals of Photonics
Photoconductive
material
Photo-addressed spatial light modulator
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CHAPTER 6-1---ELECTRO-OPTICS
Dichroic reflector
of red light
Polarizing
beamsplitter
Transparent
electrodes
White light
(blue)
Modulated
light
BSO
Incident read light
(red)
Pockels readout optical modulator (PROM).
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CHAPTER 6-1---ELECTRO-OPTICS
Electro-optics of Liquid Crystal
• Electrical Properties of Nematic Liquid Crystals
Anisotropic
Uniaxial symmetry
Optics axis rotate
|| (n||, ne )
z
E
 (n, no )
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CHAPTER 6-1---ELECTRO-OPTICS
Liquid crystal cell
x
x
d

E
z
z
y
y
  0,V  Vc
V  Vc

1
   2 tan exp(
), V  Vc
2
1
cos 2  sin 2 

 2
2
2
n ( )
ne
no
V0
Phase modulator
Fundamentals of Photonics
  2 [n( )  n0 ]d / 0
 max  2 (ne  n0 )d / 0
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CHAPTER 6-1---ELECTRO-OPTICS
90
80
70
60

Dependence of the tilt angle
q on the normalized rms
voltage
50
40
30
20
10
0
0
0.5
1
1.5
2
2.5
3
3.5
4
(V-Vc)/Vo
1
0.9
0.8
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.
0.7

 max
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5
3
3.5
4
(V-Vc)/Vo
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CHAPTER 6-1---ELECTRO-OPTICS
Nematic Liquid-Crystal Retarders and Modulators
Phase modulator
Polarization modulator
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CHAPTER 6-1---ELECTRO-OPTICS
s
Polarizer
x
Incident
light
Mirror
Reflected
light
Liquidcrystal
cell
y
Reflective light intensity modulator
=/2
= 0
Fundamentals of Photonics
(off state)
(on state)
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CHAPTER 6-1---ELECTRO-OPTICS
Twisted Nematic Liquid-Crystal Modulators
Linear polarization direction rotate with liquid crystal twist direction
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CHAPTER 6-1---ELECTRO-OPTICS
s
x
(a)
Bright
y
Polarizer
Polarizer
s
x
(b)
Dark
y
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CHAPTER 6-1---ELECTRO-OPTICS
s
Polarizer
Mirror
Liquidcrystal
cell
Reflective twist nematic liquid crystal modulator,
normally 45degree twisted
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CHAPTER 6-1---ELECTRO-OPTICS
Ferroelectric Liquid Crystals
Faster response (us, nematic: ms)
smectic-C phase
x
x


90°




z
z
Smectic
layers
y
y
Surface stable Ferroelectric liquid crystal (SSFLC), only on-off state
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CHAPTER 6-1---ELECTRO-OPTICS
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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CHAPTER 6-1---ELECTRO-OPTICS
Optically Addressed Spatial Light Modulators
Light-locking Dielectric
layer
mirror
Transparent
electrodes
Polarizing
beamsplitter
Modulated
light
White light
Incident read light
(red)
Photoconductor
Liquid crystal
Hughes liquid-crystal light valve
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CHAPTER 6-1---ELECTRO-OPTICS
Photorefractive materials
light
free charge carriers
space-charge distribution
refractive index distribution
(b)
Conduction band
(c)
(a)
LiNbO3
Fe3+
Fe2+
Valence band
(d)
x
Electric
field
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CHAPTER 6-1---ELECTRO-OPTICS
Simplified theory of photorefractivity
rate of photoionization G ( x)  s( N D  N D ) I ( x)
ND : the number density of donors
ND+: the number density of ionized donors
S : the photoionization cross section.
electrons recombination rate
R( x)   R n( x) N D
n(x): electrons density , R is a constant
In equilibrium, R(x) = G(x),
sI ( x)( N D  N )   R n( x) N

D
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
D
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CHAPTER 6-1---ELECTRO-OPTICS
s N D  N D
n( x ) 
I ( x)

 R ND
Electric Field J  ee n( x) E ( x)  k BT e
dn
0
dx
e : electron mobility
K: Boltzmann’s constant
T : temperature.
k BT 1 dn
E ( x) 
e n( x) dx
Refractive Index
Fundamentals of Photonics
1 3
n( x)   n rE ( x)
2
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CHAPTER 6-1---ELECTRO-OPTICS
E ( x) 
k BT 1 dI
e I ( x) dx
1 3 k BT 1 dI
n( x)   n r
2
e I ( x) dx
EXAMPLE
2 x
Incident light I ( x)  I 0 (1  m cos
)

 sin(2 x / )
sin(2 x / )
E ( x)  Emax
, n( x)  nmax
1  m cos(2 x / )
1  m cos(2 x / )
If m is
small
2 x
n( x)  nmax sin

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CHAPTER 6-1---ELECTRO-OPTICS
Nonuniform
light
Nonuniform
light
I(x)
Photoionization
Diffusion
x
Freecarrier
density
x
Recombination at
traps
Fixedcharge
density
E(x)
Electric field
Refractive index
grating
+
+
-
x
x
△n(x)
x
Response of a photorefractive material to a sinusoidal spatial light pattern
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CHAPTER 6-1---ELECTRO-OPTICS
Applications of the Photorefractive Effect
Wave 1 (reference)
Grating
Two-wave mixing: dynamic holography
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