chapter 12. interaction of light and sound

Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.

Chapter 12. Interaction of Light and Sound

12.0 Introduction

Acousto-Optic(AO) effect : # Effect of change in the index of refraction of medium (crystal) by an Acoustic wave# Acoustic wave Photoelastic effect Change in refractive index

Reference : A. Ghatak, K. Thyagarajam, “Optical Electronics”, Cambridge Univ. PressA. Yariv, P. Yeh, “Optical Waves in Crystals”, John Wiley & Sons


Photoelastic effect

: Mechanical strain Index of refraction

Index ellipsoid for Principal axes : 123

2

22

2

21

2

n

z

n

y

n

x

Strain tensor elements, S :

zxxz

zyyz

yxxy

zzyyxx

Sxw

zuS

Syw

zvS

Sxv

yuS

zwSy

vSxuS

,,

where, u, v, w : displacements along the x, y, z axes

xyzxyz

zzyyxx

SSSSSS

SSSSSS

654

321

,,

,, : Normal strain

: Shear strain


Change in index of refraction due to the mechanical strain :

)6,,1()1

(6

12

iSpn j

jiji

where, Pij : Elasto-Optic (Strain Optic) Coefficient (6x6 matrix) Table 9.1 / 9.2

The equation of the index ellipsoid in the presence of a strain field :

1222

111

654

323

222

2

221

2

jjj

jjj

jjj

jjj

jjj

jjij

SpxySpxzSpyz

Spn

zSpn

ySpn

x


Example) Sound wave propagating along the z direction in water

Sound wave : )cos(ˆ),( KztzAtzw

0)/()/(,0)/()/(,0)/()/(

)sin()sin(/,0/,0/

654

321

xvyuSSxwzuSSywzvSS

KztSKztKAzwSSyvSSxuSS

xyzxyz

zzyyxx

Elasto-Optic Coefficient for the water (isotropic, Table 9.1) :

)(2

100000

0)(2

10000

00)(2

1000

000

000

000

1211

1211

1211

111212

121112

121211

pp

pp

pp

ppp

ppp

ppp

pij

0)1

(

),sin()1

(

),sin()1

()1

(

6,5,42

1132

122212

n

KztSpn

KztSpnn

The new index ellipsoid :

1)sin(1

)sin(1

)sin(1

1122

1222

1222

KztSp

nzKztSp

nyKztSp

nx


Example) y-polarized Shear wave propagating along the z direction in Ge

Sound wave : )cos(ˆ),( KztyAtzv

0)/()/(,0)/()/(

),sin()sin()/()/(

,0/,0/,0/

65

4

321

xvyuSSxwzuSS

KztSKztKAywzvSS

zwSSyvSSxuSS

xyzx

yz

zzyyxx

Elasto-Optic Coefficient for the Ge (cubic, Table 9.1) :

44

44

44

111212

121112

121211

00000

00000

00000

000

000

000

p

p

p

ppp

ppp

ppp

pij

0)1

(

),sin()1

(

,0)1

()1

()1

(

6,52

4442

322212

n

KztSpn

nnn

The new index ellipsoid :

1)sin(2)(1

44222

2 KztSyzpzyx

n


Acousto-Optic effect

Bragg diffraction & Raman-Nath diffraction

Bragg diffraction: acoustic wave vector is well defined

Raman-Nath diffraction: acoustic wave vector has an angular distribution

# Spread angle of Acoustic wave :

nB 2~

L

~

# Diffraction angle of Light :

L

acoustic wave

light wave

# Dimensionless parameter :2

24

n

LQ B

ndiffractioNath -Raman

ndiffractio Bragg

: 1

: 1

Vector Representation


Raman-Nath diffraction :Multiple order diffraction

Bragg diffraction :Single order diffraction

Example) Water, n=1.33, =6MHz (vs=1,500 m/s), =632.8 nm

m250/ sv

Regime Bragg

RegimeNath -Raman

: cm 2

: cm 2)2/(2

nL


Raman-Nath diffraction

Moving periodic refractive index grating : Kztnntzn sin, 0

Consider L is small enough so that the medium behave as a thin phase grating,

)sin(,2

10 KztLtzn

where, n0L, 1=(2/)n0L

The transmitted field on the plane x=L :

)]sin([0

10 Kzttit eEE

im

mm

i eJe

)(sin

)(

]}){(}){()([

where,

22121110

)(0

0

Kzt

eeJeeJJeEE iiiitit


])[(

])[(

)(

]2)2(]2)2(120

])(])(110

)(100

00

00

0

kztikzti

kztikzti

tit

eeJE

eeJE

eJEE])([

1000 0)( Lxktit eJEE

amplitude reduction

Frequency : ,Wave vector :

2/12221

2/12221

]/)[(

,]/)[(

kck

kck

])()[(110

])()[(110

01

01

)(

)(

kzLxkti

kzLxkti

eJE

eJE

Propagation in x>L :

: +1 order

: -1 order

nk

Knk

K

1

1

sin

sin

Diffraction angles : 1

1


m-th order diffractive wave : # Frequency : # Diffraction angle :

m

nmm

sin

,654.8,520.5,405.22

,0)( 110 nLJ

582.0)(85.1 111 J : First order diffraction maximum

!12 2

2

L#

: The restriction on length of medium is severe at higher frequency Diffraction efficiency reduction


Bragg diffraction

In this regime, we can no longer consider the refractive index perturbation to act as a thinphase grating. We should consider the propagation equation of light ;

e)sin(e2

2

02 kzt

t

2

2

02

2

02 )sin()(

t

ekzte

t

2

2)(

02

2

02

2

2

2

)(2

1 )(

t

eee

it

e

z

e

x

e kztikzti

: light) : acoustic, :( problem D-2 toconverting z x

eeee 0

: e-field Total

where,])[)(

)(0

)(00

),(),(

),(),(zxtiti

zxtiti

ezxAezxAe

ezxAezxAe

rk

rk


222

02

2220

2

)(

k

k

z

Ak

z

A

x

Ak

x

A

2

2

2

2

,

Let,

And, slow varying approximation ;

)])[(2)])[(2)][(0

)()(0

)])[(

)])[(

)(00

)()(2

1

2

2

2

zxtizxtizxtikztikzti

zxti

zxti

zxti

eAeAeAeei

ez

A

x

Ai

ez

A

x

Ai

ez

A

x

Ai


])([00

2)(

])([])([0

2)(00

2

12

2

12

zkxizxi

zkxizkxizxi

eAi

ez

A

x

Ai

eAeAi

ez

A

x

Ai


(1) Small Bragg angle diffraction

00

z

A

])([00

2

)(

])([])([0

2

)(0

2

1

2

2

1

2

zkxi

zxi

zKxizKxi

zxi

eAi

edx

dAi

eAeAi

edx

dAi

KK & : Bragg condition


AA

AA

eAkdx

Ad

eAkdx

Ad

ix

ix

2/1

0

0

2/1

00

2/10

2

0

0

2~

2~

)(4 where,

~~

~~

These equations have a solution for when only The solutions for +=, -= are independent each other, so let

A

0~

~)(

~

020

20

2

Adx

Adi

dx

Ad

2/12412

)(0

)(00

)(

)(~

:solution

where,

21

21

k

eDeCxA ixix

02

1

021

)()(

,

)(~

where

21

21

DiD

CiC

eeDeCxA ixixix


Diffraction efficiency,

41

21

041

21

0

0

,

condition) (initial0)0(~

,1)0(~

DC

xAxA

)(sin)(

~)(

)(sin2)(cos)(~

)(

222

22

22

00

xxAxP

xxxAxP

0-th and 1-st diffraction powers :

i) 24122

0 )(1)()( xPxP

ii) Maximum transfer : ;0

)(sin)(

)(cos)(2

20

xxP

xxP

Diffration efficiency :

,23,21

)(sin)( 2

L

LLp


20

11

nexpressionscalar :4

0 Spn

B

B

nc

Spn

c

cos

cos4

3

23

2

1SvI aa Acoustic intensity : (Text p. 483)

2/12 )(

cos2a

B

IM

3262 / where, avpnM : Figure of Merit

Diffraction efficiency :

LIM a

B

2/12

2 )(cos2

sin


Acoustic intensity for maximum

efficiency :2

2

22

2

cos

LMI B

am

Acoustic power for maximum efficiency (LH cross-section, maximum impedance matching case) :

22

22 1

2

cos

ML

H

MLHIp B

aam

Diffraction figure of merit of the material relative to water


(2) Large Bragg angle diffraction

02/

x

A

])([

002)(

])([])([0

2)(0

4

14

1

zKxizxi

zKxizKxizxi

eAez

A

eAeAez

A

x-dependent term


These equations have a solution for when only The two solutions are independent each other, so let

1) Co-directional coupling)

A K

zi

zi

eAz

A

eAz

A

)(0

)(0

~~

~~

k

AA

AA

c

2/1

0

0

2/1

00

2/10

2

2

2/10

2

2~

2~

)(

1

4)(4 where, 2/12

412

)(0

)(00

)( ,

)(~

where

21

21

iziz eDeCzA

Solutions :

02

1

021

)()(

)(~

where,

21

21

DiD

CiC

eeDeCzA iziziz


Diffraction efficiency,

0)0(~

,1)0(~

0 zAzA

41

21

041

21

0 , DC

0-th and 1-st diffraction powers :

)(sin)(~

)(

)(sin2)(cos)(~

)(

222

22

22

00

zzAxP

zzzAzP

Diffraction efficiency :

}1{sin

4/1

1

)(sin

2/1

4

2

22

22

2

2

L

L


2) Counter-directional coupling)

zi

zi

eAz

A

eAz

A

)(0

)(0

~~

~~

K where,

2/12412

21212/)(

)(011

)(0

)(

1)(

~

)(~

where,

21

g

egQegPezA

eDeQePezA

gzigzizi

izgzgzzi

Solutions :


0)(

~,1)0(

~0 DzAzA

gDgDg

gDgADP

gDgDg

gADP

222122

222122

222122

22

00

sinhcosh

sinh~)(

sinhcosh

~)(

# Application : DBR reflector


Surface Acousto-Optics

: Diffraction effect through a thin film surface or wave guide: High intensity localized on the interface enhancing the diffraction efficiency : 1967, Ippen et al. (first experimental demonstration in a quartz)

Surface undulation profile by the acoustic wave :

2,sin KzKax

zsinxcosk

zsinxcosk

ttt

rrr

nk

k

Wave vectors of reflected and transmitted light waves :

Electric field on the surface :

zxik sincosexpEE 0


1) Reflection wave

dzdeEec r-ikrikr

rrE

l

ill

i

zki

eJe

dzdecE

)(

E

sin

rik)sin()cos(cosxik0r

rr

deklKJcE

ka

dzdeJcE

ll

r

l

zzilKl

rik0

rik)i(ksin0r

r

r

)sin(2)(

)cos(cos

)(E

where,


,2,1,0

)sin(sinsinsin

)(2E

where,

)sin(cos0r

r

l

lKklKkk

eJcE

rr

l

zlkkixikl

zkx rrr sinkcosrk

Diffraction angle :

l

sinsin r

,cos2r ka the amplitude of reflected wave :

l

zlkkixikl ekaJcE )sin(cos

0r )cos2(2E


2EE 0, If 0r

rcra

l

lil

l

zlkkixikl

ekaJr

ekaJrE

r)zkk(0

)sin(cos0r

)cos2(E

)cos2(E

Diffraction efficiency of l-th order :

)cos2(22 karJr ll


2) Transmitted wave

Similarly,

lKknk

enkakaJtE

nk

dzdeEc

t

zlKkixtl

zkin

sinsin

coscosE

sin

E

where,

where,

)sin(inkcos0t

t

rik)sin()cos(cosxik0t

t

rt

Diffraction efficiency of l-th order :

)coscos(cos

cos 22

tlt

l nkakaJtn

chapter 12. interaction of light and sound

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