EE16.468/16.568Lecture 5Electro-optics Electro-optic effect: Linear electro-optic effect: Pockels effect Kerr effect When no E applied, Refractive indices.

$Page 1: EE16.468/16.568Lecture 5Electro-optics Electro-optic effect: Linear electro-optic effect: Pockels effect Kerr effect When no E applied, Refractive indices.$
EE16.468/16.568 Lecture 5Electro-optics

Electro-optic effect:

...),( 3)3(2)2()1(0 EEEP

, D

,0 B

,t

BE

,t

DjH

EPED 0

Linear electro-optic effect:

)1( )2()1(0 kijkijij E ,1 kijk Er

Pockels effect

Kerr effect

2

2

2

0

33

22

11

0

00

0

00

z

y

x

n

n

n

When no E applied,

Refractive indices on principle axes


,22

12

2

2

2

2

2

20

2

33

2

22

2

11

2

zyx

zyxe n

z

n

y

n

x

r

DDDDU

,12

2

2

2

2

2

zyx n

z

n

y

n

x Refractive index ellipsoid

Refractive index ellipsoid:

DD

DEU e

2

1

2

1 When no electric field applied,

nx

ny

nz


,0 kijkij ErWhen electric field applied,

Refractive index ellipsoid:

:ijkr 3x3x3 tensor, 27 components

,jikijk rr

DD

DEU e

2

1

2

1 Generally,

,1,

jijiij xx is a 3x3 tensor, = -1 or = -1

Why?

x

y

zx

y

z

y

x

z

No EO effect for crystal with central symmetry, why?

x

y

z+

-,zjizzijz ErEr


,01 kijk Er

,jikijk rr

j i = 1 2 3

1

2

3

1 6 5

6 2 4

5 4 3

3

2

1

636261

535251

434241

333231

232221

131211

0

E

E

E

rrr

rrr

rrr

rrr

rrr

rrr


When Ez only,

3

636261

535251

434241

333231

232221

131211

0 0

0

E

rrr

rrr

rrr

rrr

rrr

rrr


Take LiNbO3 as an example :

,1111 2

3320

2132

0

2132

0

zEr

nyEr

nxEr

n zzz

,11

3320

2

z

e

Ernn

,2

133

300 ze Ernnn

,2

113

300 zErnn

3330

3

22

51

51

33

1322

1322

0 0

0

00

00

00

00

0

0

Er

E

r

r

r

r

rr

rr

,1

'

1132

020

zEr

nn

,2

133

30 ze Ernn


Take LiNbO3 as an example:

,2

133

30 zErnn

Z cut

TM polarization

TE polarization

,2

113

300 zErnn


,2

133

300 ze Ernnn ,

2

133

30 zErnn

,2

113

300 zoo Ernnn ,

2

113

30 zErnn

Different index changes of TE and TM waves


Take LiNbO3 Intensity modulator Z cutTM polarization

,21 AAAout ,2

10

01LnieAA ,

2

1 )(02

0 LnnieAA

),1(2

10

0nLiLni

out eeAA ),2/(cos)1(4

1 20

2

0

2nLIeIAI nLi

outout

),)2

1((cos)(cos 33

30

20

20 L

d

VnInLIIout

,2/)2

1( 33

30

L

d

Vn

Ln

dV

3330


Take LiNbO3 Intensity modulator Z cutTM polarization

),)2

1((cos)(cos 33

30

20

20 L

d

VnInLIIout

Frequency double region

Linear region


Operating characteristics

,max

minmax

I

II

• Insertion loss (dB):

• Modulation depth

• Bandwidth: the highest frequency the modulator can operate, R and C

%,100 0min I

),log(10)(in

out

I

IdBIL


for a channel waveguide, assuming the E filed is uniform,

• Power consumption: P/f, f: bandwidth

,)(2

1 2 dVEffWPe

),(2

)(2

1 22 ELdW

dVEW d Take LiNbO3 as an example:

LndVE

3330

/

,)(2

)(2

1 2

3330

2

nL

dWdVEW d P/f ~ 2µW/MHz

d

LW

d

AC d

C ~ 0.4pF


for a bulk EO modulators, assuming the E filed is uniform,

• Power consumption: P/f, f: bandwidth

,)(2

1 2 dVEffWPe

),(2

)(2

1 22 ELdW

dVEW d

Take LiNbO3 as an example:Ln

dVE33

30

/

,)(2

)(2

1 2

3330

2

nL

dWdVEW d P/f ~ 2W/MHz

z ,2

133

30 zErnn

d

LW

d

AC d

C ~ 3pF


Phase modulator:

,2

2

1

033

30 LErn z

,2

133

30 zErnn Phase shift due to the applied voltage:

,33

30

0

rnE

L

dV z

Examples:


+ -

z

,2

113

30 zErnn

z ,2

133

30 zErnn


Polarization modulator:

,2

122

30 yx Ernn

Phase shift due to the applied voltage:

,2 33

30

0

rnE

L

dV

0

0

00

00

00

00

0

0

2

22

51

51

33

1322

1322

0 E

r

r

r

r

rr

rr

x

y

,2

122

30 yy Ernn


Polarization modulator:

,2

122

30 yx Ernn

Phase difference due to the applied voltage:

,/223022

30 WVrLnErLn yy

x

y

,2

122

30 yy Ernn Ay

Ax

,sin0, AA in

,cos0, AA iny

),2

1(exp(sin)exp(sin 22

30000, yxout ErnnLiALniAA

),2

1(exp(cos)exp(cos 22

30000, yyouty ErnnLiALniAA

W


Polarization modulator: Phase difference due to the applied voltage:

,/223022

30 WVrLnErLn yy

x

y

Ay

Ax

W

When = /2,

Ay

Ax

Ay

Ax

When = ,

Ay

Ax


,/2

1 30 dVnn

0

30

60

90

120

150

180

210

240

0 1 2 3 4

L = 2mm

L = 5mm

Bias (V)

Pha

se S

hift

(de

gree

)

TE-TM converter:


Quantum Key Distribution (QKD)

JUSTIN MULLINS, IEEE SPECTRUM, p. 40, May 2002

Single photon

Bennett and Brassard 1984 protocol Proc. IEEE Int. Conf. Computers, Systems and Signal Processing, 1984, pp. 175–179.



JUSTIN MULLINS, IEEE SPECTRUM, p. 40, May 2002

Bennett and Brassard 1984 protocol Proc. IEEE Int. Conf. Computers, Systems and Signal Processing, 1984, pp. 175–179.



W. T. Buttler, etal. Physical Review A, Vol 57, 1998, pp. 2379-2382


Integrated QKD Receiver

Input photon polarization

states

Detection polarization

basis

Final polarization states

Si single photon detector

signal

TE Linear TE 0

TE Circular TE & TM x

TM Linear TM 1

TM Circular TE & TM x

Circular – left-hand

Linear TE & TM x

Circular – left-hand

Circular TE 0

Circular – right-hand

Linear TE & TM x

Circular – right-hand

Circular TM 1

Transmitter: Alice

Receiver: Bob

QKD Links

QKD Links

Receiver: Bob


,/2

1 3 dVnn

0

30

60

90

120

150

180

210

240

0 1 2 3 4

L = 2mm

L = 5mm

Bias (V)

Pha

se S

hift

(de

gree

)


Input photon polarization states

Applied voltage

Detection polarization basis

Final polarization states

detected signal

TE 0 Linear TE 0

TE V Circular Circular x

TM 0 Linear TM 1

TM V Circular Circular x

Circular – left-hand 0 Linear Circular x

Circular – left-hand V Circular TE 0

Circular – right-hand 0 Linear Circular x

Circular – right-hand V Circular TM 1


Poling of polymer materials, contact poling and corona poling:

EO Polymer poling


• Lower voltage ~ 800V

• Good film quality

• Easy control of poling voltage

Advantages:

• Uniform poling voltage

Heating/Cooling block

V

Buffer Layer

Film

Poling Electrode

• Select poling area

Heating/Cooling blockBuffer Layer

EO polymer

ElectrodeNeedle

V


I-V curve during poling:

• Poling voltage : 900V

• EO coefficient ~ 22pm/V


TM-pass waveguide Polarizer

TM input TE input

Before poling After poling

n (TE) 1.594 1.591

n (TM) 1.594 1.598


Directional coupler:

1 3

Un-connected port

A1(z)

A2(z)

For A1(0) = A1, A2(0) =0

4

ziezAidz

zdA )()(

21

ziezAidz

zdA )()(

12

12

zizi ezAzAdz

di

dz

zdAe

dz

d

)()(

)(1

22

1

0)()()(

121

21

2

zAdz

zdAi

dz

zAd

0)()()(

222

22

2

zAdz

zdAi

dz

zAd



1 3

Un-connected port

A1(z)

A2(z)

For A1(0) = A1, A2(0) =0

4

0)()()(

121

21

2

zAdz

zdAi

dz

zAd

0)()()(

222

22

2

zAdz

zdAi

dz

zAd

))2/(sin())2/(cos()( 222221 ziBzAezA

zi

))2/(sin())2/(cos()( 222222 ziDzCezA

zi


1 3

Un-connected port

A1(z)

A2(z)

For A1(0) = A1, A2(0) =0

4

))2/(sin())2/(cos()( 222221 ziBzAezA

zi

))2/(sin())2/(cos()( 222222 ziDzCezA

zi

))2/(sin())2/(cos()0()( 222221

21 ziBezAezA

zi

zi

))2/(sin()( 2222 ziDezA

zi

)(

))2/(sin(2

))2/(cos()2/()(

1

222222222

zAi

zeDziDezAd

d zi

zi

22

1

4)(

)0(

AD

22

1

4)(

)0(2

A

B



1 3

Un-connected port

A1(z)

A2(z)

)cos()0()( 11 zAzA

)sin()0()( 12 ziAzA

For A1(0) = A1, A2(0) =0

4

LPP

2

222

2

2

02 2sin

2

/

),2

1(

2233

30 zErnn

))2/(sin()2/(

)0()( 22

22

122 z

AiezA

zi


For A1(0)= A2(0) = ½ A(0),

))2/(sin())2/(cos()( 222221 ziBzAezA

zi

))2/(sin())2/(cos()( 222222 ziDzCezA

zi

))2/(sin())2/(cos()0()( 22221

21 ziBzAezA

zi

))2/(sin())2/(cos()0()( 22222

22 ziDzAezA

zi


))2/(sin())2/(cos()0()( 22221

21 ziBzAezA

zi

))2/(sin())2/(cos()0(

2

))2/(cos()2/())2/(sin()2/()0()(

22221

2

222222221

21

ziBzAei

ziBzAezAdz

d

zi

zi

))2/(sin())2/(cos()0()()( 2222

22

21 ziDzAeiezAidz

zdA zi

zi

)0(2

)2/()0( 122

2 Ai

iBAi

)0()2/(2

222AB


))2/(sin())2/(cos()0(

2

))2/(cos()2/())2/(sin()2/()0()(

22221

2

222222221

21

ziBzAei

ziBzAezAdz

d

zi

zi

))2/(sin())2/(cos()0()()( 2222

22

21 ziDzAeiezAidz

zdA zi

zi

)0()2/(2

122AD

))2/(sin())2/(cos()0()( 22221

21 ziBzAezA

zi

))2/(sin())2/(cos()0()( 22222

22 ziDzAezA

zi

)0()2/(2

122AB


)0()2/(2

122AD

))2/(sin())2/(cos()0()( 22221

21 ziBzAezA

zi

))2/(sin())2/(cos()0()( 22222

22 ziDzAezA

zi

)0()2/(2

122AB


)0()2/(2

122AD

))2/(sin())2/(cos()0()( 22221

21 ziBzAezA

zi

))2/(sin())2/(cos()0()( 22222

22 ziDzAezA

zi

)0()2/(2

122AB

When = 0,

)sin()cos()0()( 11 zizAzA

)sin()cos()0()( 12 zizAzA


When 0,

))2/((sin)2/(2))2/((cos)0()( 222

2

22

22211 zzPzP

))2/((sin)2/(2))2/((cos)0()( 222

2

22

22222 zzPzP


Traveling-wave electrodes: 1 3

Un-connected port

A1(z)

A2(z)

)cos()0()( 11 zAzA

)sin()0()( 12 ziAzA

For A1(0) = A1, A2(0) =0

4

For high-speed electrode using transmission lines:

Vg(t) VL

A

z = 0

B

l

ZL

z = - l

Z0Vi

-V0

+V0=

ZL Z0-

ZL Z0+


Standing waves for impedance mismatching:

Vg(t) VL

A

z = 0

B

l

ZL

z = - l

Z0Vi

-V0

+V0=

ZL Z0-

ZL Z0+

|V(z)|

2|V0| +

- -3/4 -/2 -/4

Standing waves

,2

133

30 zErnn

Cancelled


Traveling waves for impedance matching:

Vg(t) VL

A

z = 0

B

l

ZL

z = - l

Z0Vi

-V0

+V0=

ZL Z0-

ZL Z0+

traveling waves

,2

133

30 zErnn

|V(z)|

|V0| +

- -3/4 -/2 -/4


Kerr effect :


Applications of Kerr effect :

• Self-phase modulation

,22 InIKEn

• Self-Focusing


Acoustic-optic Modulators• Bragg condition:

,2

sin2

2

),2

cos()( 00 xtnnxn

,2

1

sIn

,2sInR


Doppler shift

Angle mismatch


Acoustic –optics devices

• Modulator

• Beam Scanner

Number of resolvable spots:

Bandwidth: ,22

sin fvs

,D

Bvs


Acoustic –optics devices • Free space inter-connector

• Isolator


Light propagation in anisotropic crystals

Optical axises

no

o light



Optical axis

no

o light

ne



Optical axis

no

ne

Polarization beam splitter





ne no


Faraday rotator


Fiber–optics devices • Coupler FC/APC

SC/PC

FC/PC

SC/APC


Circulator

EE16.468/16.568Lecture 5Electro-optics Electro-optic effect: Linear electro-optic effect: Pockels effect Kerr effect When no E applied, Refractive indices.

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