chapter 19. acousto optics - hanyangoptics.hanyang.ac.kr/~choh/degree/[2014-1] photonics...nonlinear...

Nonlinear Optics Lab. Hanyang Univ.

Chapter 19. Acousto-Optics

19.1 Interaction of Light and Sound

19.2 Acousto-Optic Devices

19.3 Acousto-Optics of Anisotropic Media


Acousto-Optic effect : The refractive index of an optical medium is altered by the presence of sound.

Light can be controlled by sound~

Applications : Optical Modulators, Switches, Filters, Isolators,

Frequency shifters, Spectrum analyzers, …

Change in refractive index of material by sound wave :

- Gas : Density modification by dynamic strain(sound wave)

- Solid, Liquid : Alternation of the optical polarizability by

the vibrations of molecules


An acoustic wave creates a perturbation of the refractive index in the form of a wave.

Therefore, the medium becomes a “dynamic” graded-index medium (an inhomogeneous

medium with a time-varying stratified refractive index).

The theory of acousto-optics deals with the perturbation of the refractive index, and with the

propagation through the perturbed time-varying inhomogeneous medium.

However, since optical frequencies are much greater than acoustic frequencies. As a

consequence, it is possible to use an adiabatic approach in which the optical propagation

problem is solved separately at every instant of time, always treating the medium as if it were

a static inhomogeneous medium. In this quasi-stationary approximation, acousto-optics

becomes the optics of an inhomogeneous medium (usually periodic) that is controlled by

sound.


19. 1 Interaction of light and sound

A. Bragg Diffraction

Considering an acoustic plane wave traveling in the x direction, the strain is

where, : angular frequency, : wavenumber

Acoustic intensity:

Refractive index change is given by [analogous to the Pockels effect in (20.1-4)]

where, : Photoelastic constant (strain-optic coefficient)


Substituting from (19.1-2) into (19.1-5),

sI (19.1-6)

where, : Figure of merit for the strength of the AO effect in the material

Consequently, refractive index of medium is

where,


Figure of merit of the material

relative to water


Consider an optical plane wave traveling through the acousto-optic medium,

Refractive index can be regarded as a static “frozen” sinusoidal function

(Adiabatic approach),

where, 𝜑 = Ω𝑡 : fixed phase


Amplitude reflection

Divide the medium into incremental planar layers orthogonal to the x axis, since the optical

plane wave is partially reflected at each layer because of the refractive-index change.

And, assume that the reflectance is sufficiently small (transmitted light is not depleted),

Total complex amplitude reflectance :

Included since the reflected wave at x is advanced by a distance 2xsinq, corresponding to a

phase shift 2kxsinq, relative to the reflected wave at x=0. (refer to p.178, “Intro. to Optics”, Pedrotti)

Independent of x


][)sin( )qx(j)qx(j eeqx Using complex notation,

where,

Performing the integral, and substituting 𝜑 = Ω𝑡,

where,

-r ,r: upshifted and downshifted reflections, respectively

L is sufficiently large, two maxima at 𝜃 = ±sin−1(𝑞/2𝑘) are sharp, so that any slight

deviation from the angles makes the counter term negligible. Thus, only one of these

two term is significant at a time.


Bragg condition (upshifted reflection)

Sinc function has maximum when the argument is zero: 0sin 2 qk q (for upshifted reflection)

: Bragg angle is the angle for which the incremental reflection from planes

separated by an acoustic wavelength L have a phase shift of 2p

(2𝑘𝑥 sin 𝜃 = 2𝑘Λ ∙ 𝜆/2Λ = 2𝜋) so that they interfere constructively.


Bragg condition (𝑞 = 2𝑘𝑥 sin 𝜃𝐵) is equivalent to the vector relation,


Tolerance in the Bragg condition

(19.1-14)

])2sinsinc[(sin

]2)sin(sin2sinc[]2)sin2sinc[(

qq

pqqpq

/L

/Lk/Lqk

B

B

qk B qsin 2

First zero at .

Since L is typically much greater than . This is an extremely small angular width.

L/B 2sin sin qq L/B 2qq


Doppler shift

(19.1-14) )exp( tjr

Incident light field is proportional to [ ],

so the reflected wave has the form of ,

and has angular frequency

)exp( tjE )exp( tj

])(exp[ tjErEr

: The frequency as well as the propagation direction of the reflected wave is changed.

The frequency shift equal to the frequency of sound.

This can almost be thought as a Doppler shift.

q q

r vs

vssinq

Doppler shifted frequency :

LL pp )]2)(2)(2(21[ ///r


Peak reflectance (skip)

Reflectance at the Bragg angle q=qB is [from (19.1-12)]

From (6.2-8), (6.2-9),

n1→n+Dn, n2=n, q1=90o-q, n1sinq1=n2sinq2, and neglect term of 2nd order in Dn,

(TE)

(TM) For small q, cos2q ~1

Report)


From (19.1-18), by using ,

or, by (19.1-6)

sinq=/2L

- R : : typical light-scattering phenomena. 44

0

- R : valid only low intensity region since this result

is obtained from the approximation theory

(19.1B : Coupled-wave theory)

sI


Downshifted Bragg Diffraction

qk B qsin 2


Quantum interpretation

)( k, )( k,r

)( q,

Photon (, k) interacts with acoustic phonon (, q),

and generates a new reflected photon (r , k)

Energy and momentum conservation conditions:


B. Coupled-wave theory

Bragg diffraction as a scattering process

Wave equation in a homogeneous medium with a slowly varying inhomogeneous

refractive-index perturbation Dn, (5.2-20)

where,

Radiation Source Term


First Born approximation :

Assume that the scattering source S is induce by the incident field

[not by the actual (local) field].

- Incident field (plane wave),

- Perturbation in n caused by acoustic wave (plane wave),


Coupled-wave equations (upshifted Bragg diffraction)

where,

(19.1-28)

Refer to Appendix II ~…


Small-angle Bragg diffraction

For the case of small-angle diffraction(q<<1), two waves travel approximately in the z direction

(19.1-35)

where,

Boundary condition : 00 )(Ar


Reflectance :

D

0

02sin

p dn2

0

0

D

p dn

d


C. Bragg diffraction of beams

Diffraction of an optical beam from an acoustic plane wave

For an optical beam of width D interacting with an acoustic plane wave, the optical beam

can be decomposed into plane waves with directions occupying a cone of half-angle,

Rectangular: 1𝜆

𝐷, Circular: 1.22

𝜆

𝐷,

Gaussian(waist diameter=2w0 ): 𝜆

𝜋𝑤0= 0.64

𝜆

𝐷

Although there is only one wave-vector

q, there are many wave-vectors k within

a cone angle. But there is only one

direction k for which the Bragg

condition is satisfied.

The reflected wave is plane wave

with only one wave-vector kr.


Diffraction of an optical beam from an acoustic beam

Suppose now that the acoustic wave itself is a beam of width Ds. Then angular divergence,

If the acoustic beam divergence is greater than the optical beam divergence (𝛿𝜃𝑠 ≫ 𝛿𝜃),

and if the central direction of two beams satisfy the Bragg condition, every optical plane

wave find acoustic match and the reflected light beam has the same divergence as the

incident beam.


Diffraction of an optical plane wave from a thin acoustic beam :

Raman-Nath Diffraction


Thin acoustic beam act as a thin phase (diffraction) grating.

Quantum interpretation :

One incident photon combines with two phonons to form a photon of second-order

reflected wave. Conservation of momentum condition is 𝑘𝑟 = 𝑘 ± 2𝑞. The energy

conservation condition is 𝜔𝑟 = 𝜔 ± 2Ω. Similar interpretations apply to higher orders.

Raman-Nath diffraction

(Debye-Sears scattering)

Refer to Appendix III~ multiple order diffraction


19. 2 Acousto-optic devices

A. Modulators

Using an electrically controlled acoustic transducer (PZT), the intensity of the

reflected light can be varied. can be used as a modulator or switch.

D

0

02sin

p dnRe

(19.1.B), Reflectance, can be unity : 100% reflection

Efficient frequency shifter


Modulation bandwidth

Bandwidth : Maximum frequency range(band) in which modulator can efficiently modulate.

[In idealized condition]

When both the optical and the acoustic waves are

plane waves, the frequency of sound corresponds

to a Bragg angle,

For a fixed angle of incidence q, an incident monochromatic optical plane wave of

wavelength interacts with one and only one acoustic wave frequency f satisfying (19.2-1).

The frequency of reflected wave is +f.

Although the acoustic wave is modulated, the reflected optical wave is not. That is, under

this idealized condition the bandwidth of the modulation is zero!


Bandwidth : or : Transit time of sound wave

Necessary time for the sound wave

to interact at all point The light beam should be more tightly focused

in order to increase the bandwidth.

[ Modulation with a bandwidth B]

The sound wave is still planar, but the incident light is a beam of width D, and then

angular divergence 𝛿𝜃 = 𝜆/𝐷.


B. Scanners

Relation between the angle of deflection and the sound frequency :

By changing sound frequency, the deflection angle can be varied. But both the angle of

incidence and the sound frequency must be changed simultaneously.

Tilting the sound beam!

The angle of tilt must be

synchronized with the

acoustic frequency f~


[Another method instead of tilting]

To use a sound beam with an angular divergence equal to or greater than the entire range of

directions to be scanned.

As the sound wave frequency is changed, the Bragg angle is altered and incoming light wave

selects only one acoustic plane-wave with the matching direction. Efficiency is low~


Scan angle

The incident light wave interacts with the sound wave of frequency f at an angle

𝜃 = (𝜆/2𝑣𝑠 ) 𝑓, and is deflected by an angle 2𝜃 = (𝜆/𝑣𝑠 ) 𝑓.

Scan angle of deflection to cover the bandwidth B is

The divergence of sound beam should be equal or greater than this angle, that is

Larger scan angles are obtained by use of materials in which the sound velocity is smaller~


Number of resolvable spots

Number of resolvable spots :

Number of nonoverlapping angular widths of light within the scanning range

Angular width of the optical wave : 𝛿𝜃 = 𝜆/𝐷

Assuming that 𝛿𝜃 ≪ 𝛿𝜃𝑠 , the number of resolvable spots is

or


The acousto-optics scanner as a spectrum analyzer

Sound waves with different frequencies are deflected in different directions. By this,

the spectrum of sound wave can be analyzed. Spectrum analyzer


C. Space switches

Space switch : transfers information carried by one or more optical beams to one or more

selected directions.

Routing an optical beam to one of N directions by applying acoustic wave of frequencies fi.

Routing an optical beam to M directions simultaneously by applying acoustic wave of

frequencies fi simultaneously.


Routing each of two(N) optical beams to a set of specified directions simultaneously by

applying acoustic wave of frequencies fi successively.

Each pulse duration : T/N

where, T=W/vs : transit time

Spatial light modulator(SLM)

Acoustic wave frequency f is the

same, but with different

amplitudes in different segments.


Random access switch that routing each of L optical beams to M directions simultaneously

by applying acoustic wave of frequencies fi successively.

Acoustic cell is divides into

L segments, each of which

carries an acoustic wave

with M frequencies.

Interconnection capacity If an acoustic cell is used to route each of L incoming optical beams to a maximum of M

directions simultaneously, then product ML cannot exceed the time-bandwidth product N=TB.


D. Filters, frequency shifters, and isolators

Tunable acousto-optic filters Bragg condition, sin𝜃 = 𝜆/2Λ. Optical wavelength is the function of q and L(f).

Therefore, an optical wave with a wavelength can be selected successively by

adjusting the angle q (or the sound frequency f ).

Frequency shifters Bragg diffracted light is frequency shifted (up or down) by the frequency of sound, (𝜈 ± 𝑓).

Applications : Optical heterodyning (mixing two optical beams with different frequencies to

detect the beat signal), optical FM modulators, and laser Doppler velocimeters, …

Optical isolators

The frequency of returning light differs from that of

original light by twice the sound frequency.

A filter may be used to block the returning light.

Even without a filter, laser action may be insensitive

to the frequency-shifted light.


19. 3 Acousto-optics of anisotropic media

Acoustic waves in anisotropic materials

Position of molecules :

Displacement :

Strain :

- Tensile(Normal) strain (i=j) : Strain along the stress

- Shear strain (i≠j) : Strain orthogonal to the stress


The photoelastic effect (Refer to Appendix I~)

Photoeleastic effect : Change in refractive index of material by mechanical strain

In the presence of strain, the electric impermeability tensor is modified. Each of the

nine functions hij(skl) may be expanded in terms of nine variables skl in Taylor series.

where, : strain-optic (elasto-optic) tensor

Since both {hij}and {skl} are symmetrical tensors, the coefficients are invariant to

permutation of i and j, and to permutation of k and l. There are therefore only six

independent values for the set (i,j)→I and six independent values for (k,l)→K. The

fourth-rank tensor is thus described by a 6×6 matrix .

3323321331

23221221

131211

hhhhh

hhhh

hhh

Symmetrical tensor (example):


Example) Strain-optic matrix for cubic crystal (or isotropic media):

Additional constraint: There are only two independent coefficients!

(miss print !)


0

0

0

)cos(

)cos(

)cos(

0

0

0

)cos(

0

0

00000

00000

00000

000

000

000

011

012

012

0

44

44

44

111212

121112

121211

qztSp

qztSp

qztSp

qztS

p

p

p

ppp

ppp

ppp

sp ijIK

From example 19.3-1,


z


Bragg diffraction (Refer to Appendix II~)

Bragg condition:

(z-component)

(x-component)


From (19.3-18), when 𝜃𝑟 = 𝜋/2

L

0sin

cos0

q

q

nn

n

r

2/pq

2)1 /pq L

0nnr

2)2 /pq L

0nnr


Appendix

References :

A. Ghatak, K. Thyagarajam, “Optical Electronics”, Cambridge Univ. Press

A. Yariv, P. Yeh, “Optical Waves in Crystals”, John Wiley & Sons

I. Photoelastic effect

II. Bragg diffraction (Coupled-wave theory)

III. Raman-Nath diffraction


Photoelastic effect

: Mechanical strain Index of refraction

Index ellipsoid for Principal axes : 12

3

2

2

2

2

2

1

2

n

z

n

y

n

x

Strain tensor elements, S :

zxxz

zyyz

yxxy

zzyyxx

Sx

wz

uS

Sy

wz

vS

Sx

vy

uS

zwS

yvS

xuS

,,

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

xyzxyz

zzyyxx

SSSSSS

SSSSSS

654

321

,,

,, : Tensile(Normal) strain

: Shear strain

I. Photoelastic effect


Change in index of refraction due to the mechanical strain :

)6,,1()1

(6

12

D

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

32

3

2

22

2

2

12

1

2

j

jj

j

jj

j

jj

j

jj

j

jj

j

jj

SpxySpxzSpyz

Spn

zSpn

ySpn

x

Yariv, P. Yeh,

“Optical Waves in Crystals”,

John Wiley & Sons


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

D

D

DD

n

KztSpn

KztSpnn

The new index ellipsoid :

1)sin(1

)sin(1

)sin(1

112

2

122

2

122

2

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

D

D

DDD

n

KztSpn

nnn

The new index ellipsoid :

1)sin(2)(1

44

222

2 KztSyzpzyx

n


II. Bragg diffraction (Coupled-wave theory)

In this regime, we can no longer consider the refractive index perturbation to act as a thin

phase grating. We should consider the propagation equation of light ;

e)sin(e2

2

0

2 Kztt

D

2

2

02

2

0

2 )sin()(t

eKzte

t

D

2

2)()(

02

2

02

2

2

2

)(2

1

t

eee

it

e

z

e

x

e KztiKzti

D

: )dependencey (no problem D-2 toconverting

eeee 0

: e-field Total

where,

])[)(

)(

0

)(

00

),(),(

),(),(

zxtiti

zxtiti

ezxAezxAe

ezxAezxAe

rk

rk


222

0

2

222

0

2

)(

k

k

z

Ak

z

A

x

Ak

x

A

2

2

2

2

,

Let,

And, slow varying approximation ;

)]zxt)[(i)]zxt)[(i)]zxt[(i

)Kzt(i)Kzt(i

)]zxt)[(i

)]zxt)[(i

)zxt(i

eA)(eA)(eA

eei

ez

A

x

Ai

ez

A

x

Ai

ez

A

x

Ai

D

22

0

0

00

2

1

2

2

2


])([

00

2)(

])([])([

0

2)(00

2

12

2

12

zKxizxi

zKxizKxizxi

eAi

ez

A

x

Ai

eAeAi

ez

A

x

Ai

D

D


(1) Small Bragg angle diffraction

00

z

Aq

])([

00

2)(

])([])([

0

2)(0

2

12

2

12

zKxizxi

zKxizKxizxi

eAi

edx

dAi

eAeAi

edx

dAi

D

D

KK & : Bragg condition


D

D

D

D

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~

~

)(

~

0

20

2

0

2

D Adx

Adi

dx

Ad

2/12

412

)(

0

)(

00

)(

)(~

:solution

where,

21

21

D

DD

k

eDeCxAixix

021

021

)()(

,

)(~

where

21

21

DiD

CiC

eeDeCxA ixixix

D

D

DD

D


Diffraction efficiency, h

DD

41

21

041

21

0

0

,

condition) (initial0)0(~

,1)0(~

DC

xAxA

)(sin)(~

)(

)(sin2

)(cos)(~

)(

222

22

22

00

xxAxP

xxxAxP

D

0-th and 1-st diffraction powers :

i) 2

4122

0 )(1)()( D xPxP

ii) Maximum transfer : D ;0

)(sin)(

)(cos)(

2

2

0

xxP

xxP

Diffration efficiency :

,2

3,2

1

)(sin)( 2

p

ph

h

L

LLp


DD

2

0

11

nexpressionscalar :4

0 SpnD

B

B

nc

Spn

cq

q

cos

cos4

3

23

2

1SvI aa rAcoustic intensity :

2/1

2 )(cos2

a

B

IMq

p

326

2 / where, avpnM r : Figure of Merit

Diffraction efficiency :

LIM a

B

2/1

2

2 )(cos2

sinq

ph


Acoustic intensity for maximum efficiency :

2

2

22

2

cos

LMI B

am

q

Acoustic power for maximum efficiency

(LH cross-section, maximum impedance matching case) :

22

22 1

2

cos

ML

H

MLHIp B

aam

q

Diffraction figure of merit

of the material relative to water


(2) Large Bragg angle diffraction

02/

x

Apq

])([

00

2)(

])([])([

0

2)(0

4

1

4

1

zKxizxi

zKxizKxizxi

eAez

A

eAeAez

A

D

D

x-dependent term


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

1) K (Co-directional coupling)

A K

zi

zi

eAz

A

eAz

A

)(

0

)(0

~~

~~

s

s

D

D

k

AA

AA

c

D

D

D

s

2/1

0

0

2/1

00

2/1

0

2

2

2/1

0

2

2

~

2

~

)(

1

4)(4 where, 2/12

412

)(

0

)(

00

)( ,

)(~

where

21

21

D

DD iziz

eDeCzA

Solutions :

021

021

)()(

)(~

where,

21

21

DiD

CiC

eeDeCzA iziziz

s

s

D

D

DD

D


Diffraction efficiency, h

0)0(~

,1)0(~

0 zAzA

DD41

21

041

21

0 , DC

0-th and 1-st diffraction powers :

)(sin)(~

)(

)(sin2

)(cos)(~

)(

222

22

22

00

zzAxP

zzzAzP

s

D

Diffraction efficiency :

]4

1[sin4/1

1

)(sin

2/1

2

2

2

22

2

2

D

D

s

s

s

sh

L

L


2) K (Counter-directional coupling)

zi

zi

eAz

A

eAz

A

)(

0

)(0

~~

~~

s

s

D

D

KD where,

2/12

412

2121

2/)(

)(

011

)(

0

)(

1)(

~

)(~

where,

21

s

s

D

DD

D

DD

g

egQegPezA

eDeQePezA

gzigzizi

izgzgzzi

Solutions :


0)(

~,1)0(

~0 DzAzA

{ gDgDg

gDgADP

gDgDg

gADP

22

2122

22

212

2

22

2122

22

00

sinhcosh

sinh~)(

sinhcosh

~)(

D

D

D

# Application : DBR reflector


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 :

LnB

2~

q

L

L~

# Diffraction angle of Light :

L

L

acoustic wave

light wave

# Dimensionless parameter : 2

24

L

n

LQ B ppq

ndiffractioNath -Raman

ndiffractio Bragg

: 1

: 1

Vector Representation

III. Raman-Nath diffraction


Raman-Nath diffraction :

Multiple order diffraction

Bragg diffraction :

Single order diffraction

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

m250/ L sv

Regime Bragg

RegimeNath -Raman

: cm 2

: cm 2)2/(2

L pnL


Moving periodic refractive index grating : Kztnntzn D sin, 0

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

)sin(,2

10 KztLtzn D

p

where, 12p/)n0L, 12p/DnL

The transmitted field on the plane x=L :

)]sin([

010 Kztti

t eEE

qq im

m

m

i eJe

)(sin

)(

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

where,

22

121110

)(

00

Kzt

eeJeeJJeEE iiiiti

t

q

qqqq


])[(

])[(

)(

]2)2(]2)2(

120

])(])(

110

)(

100

00

00

0

KztiKzti

KztiKzti

ti

t

eeJE

eeJE

eJEE])([

100

0 0)(

Lxkti

t eJEEamplitude reduction

Frequency : ,

Wave vector :

2/1222

1

2/1222

1

]/)[(

,]/)[(

Kck

Kck

])()[(

110

])()[(

110

01

01

)(

)(

KzLxkti

KzLxkti

eJE

eJE

Propagation in x>L :

: +1 order

: -1 order

L

L

nk

K

nk

K

q

q

1

1

sin

sin

Diffraction angles : 1q

1q


m-th order diffractive wave :

# Frequency :

# Diffraction angle :

m

L

nmm

qsin

D ,654.8,520.5,405.22

,0)( 110 nLJ

p

582.0)(85.1 111 J : First order diffraction maximum (diffraction efficiency ~ 33.9%)

!12 2

2

L

pL#

: The restriction on length of medium is

severe at higher frequency

Diffraction efficiency reduction

: zero order is absent

chapter 19. acousto optics - hanyangoptics.hanyang.ac.kr/~choh/degree/[2014-1] photonics...nonlinear...

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