CHAPTER 6-1---ELECTRO-OPTICS 2015-6-9Fundamentals of Photonics 1 Chapter 6 ELECTRO-OPTICS.

CHAPTER 6-1---ELECTRO-OPTICS

23/4/21Fundamentals of Photonics 1

Chapter 6Chapter 6 ELECTRO-OPTICSELECTRO-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



Possible application

controllable focal length. Optical scanning device

U

0

Phase modulator

U

Polarization modulation

U

polarizer

analyzer

Light intensity modulator



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.



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)



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)



• 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



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



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()(



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



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



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



Longitudinal modulator Transverse modulator Traveling-wave transverse modulator

d=L several GHz

LV

V

dV

V



integrated-optical phase modulator

0

V

Input light

Modulated light

Electrodes

V

0

Cross scction

E

Waveguide



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



• 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



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



• 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



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



position switch based on electro-optic phase retardation and double refraction.

Electro-optic polarization retator

Birefringent crystal



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)



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



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



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



Pockels readout optical modulator (PROM).

BSO


Dichroic reflector of red light

White light (blue)

Incident read light (red)

Modulated light

Polarizing beamsplitter



Electro-optics of Liquid Crystal

• Electrical Properties of Nematic Liquid Crystals

AnisotropicUniaxial symmetry Optics axis rotate

|| (n||, ne )

(n, no )E

z



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



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.△



Phase modulator

Polarization modulator

Nematic Liquid-Crystal Retarders and Modulators



Reflective light intensity modulator

=/2 (off state) = 0 (on state)

Liquid- crystal

cell

sy

x

Mirror

Polarizer

Incident light

Reflected light



Twisted Nematic Liquid-Crystal Modulators

Linear polarization direction rotate with liquid crystal twist direction



x

y

Bright

sDark

x

y

s

(a)

(b)

PolarizerPolarizer



Reflective twist nematic liquid crystal modulator, normally

45degree twisted

Liquid- crystal

cell

s

Mirror

Polarizer



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°



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



Optically Addressed Spatial Light Modulators

Hughes liquid-crystal light valve


White light

Incident read light (red)

Modulated light

Polarizing beamsplitter

Light-locking layer

Dielectric mirror

Photoconductor Liquid crystal



Photorefractive materials

lightfree charge carriersspace-charge distribution

refractive index distribution

LiNbO3

Electric field

(a)

(b)

(c)

(d)

Conduction band

Valence band x

Fe3+

Fe2+



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



( ) ( )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



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



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



Applications of the Photorefractive Effect

Two-wave mixing: dynamic holography

Wave 1 (reference)

Wave 2 (object)

Grating

CHAPTER 6-1---ELECTRO-OPTICS 2015-6-9Fundamentals of Photonics 1 Chapter 6 ELECTRO-OPTICS.

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Transcript of CHAPTER 6-1---ELECTRO-OPTICS 2015-6-9Fundamentals of Photonics 1 Chapter 6 ELECTRO-OPTICS.