Nonlinear Optics Lab. Hanyang Univ. Chapter 9. Electrooptic Modulation of Laser Beams 9.0...

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(

#


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


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


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)


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


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


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


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 :


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/()/(])/([


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)


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


The ratio of the output intensity to the input :

V

VΓ

I

I

i

o

2sin

2sin 2

: amplitude modulation


Sinusoidal modulation)

tmm sin

2

t

tI

I

mm

mm

i

sinsin12

1

sin24

sin 20

tmm sin12

1

1m


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)


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


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


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


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


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

')]'(,'[)(


)/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


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

'


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

Nonlinear Optics Lab. Hanyang Univ. Chapter 9. Electrooptic Modulation of Laser Beams 9.0...

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