Ch 9-Acoustooptic Effect

8/3/2019 Ch 9-Acoustooptic Effect

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ch 9.

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Region 1 Region 2


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ch 9.

Strain waves in a material produce a periodic structure thatdiffracts light. This can be used to modulate a beam.

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ch 9.

Photoelastic effect couples mechanical strain to a changein the optical index of refraction

where Skl is the strain tensor!! ! ! ! ! and Pijkl is calledthe strain-optic tensor. Diagonal terms of S are linearstrains (volume deformation), off-diagonal terms are shearstrains (shape deformation)

The index ellipsoid in the presence of a strain is

Using contracted notation we can write the effect of strain

as

ij = 1

n2ij

= PijklSkl

(ij + PijklSkl)xixj = 1

xk = Sklxl

3 1

n2

i

= PikSk, i, k = 1, 2 . . . , 6,


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ch 9.

The strain-optic tensor Pijkl can be written incontracted notation as Pik and has a form thatcan be found from symmetry considerations for

each crystal group. The form is identical to thatof the Kerr electro-optic tensor Sik.

For example in an isotropic material

Pik =

p11 p12 p12 0 0 0p12 p11 p12 0 0 0p12 p12 p11 0 0 0

0 0 0 12

(p11 p12) 0 00 0 0 0 1

2(p11 p12) 0

0 0 0 0 0 12

(p11 p12)

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ch 9.

Consider an acoustic plane wave in waterpropagating in the z-direction

producing a strain field, !! ! ! , of

Using the Pik tensor for an isotroic medium

S3 = KA sin (tKz) Ssin (tKz)

u (z, t) = Az cos(tKz)

Sij =ui

xj

PikSi =

p11 p12 p12 0 0 0p12 p11 p12 0 0 0p12 p12 p11 0 0 0

0 0 0 12

(p11 p12) 0 00 0 0 0 1

2(p11 p12) 0

0 0 0 0 0 12

(p11 p12)

00

Ssin(tKz)000

5warnin - this is not matrix multi lication


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ch 9.

i = PikSi =

p11 p12 p12 0 0 0p12 p11 p12 0 0 0p12 p12 p11 0 0 0

0 0 0 12

(p11 p12) 0 00 0 0 0 1

2(p11 p12) 0

0 0 0 0 0 12 (p11p12)

00

Ssin(tKz)00

0

11 = 1

n21

= p12Ssin (tKz) ,

22 =

1

n2

2

= p12Ssin (tKz) ,

33 =

1

n2

3

= p11Ssin (tKz) ,

ij = 0 for i = j

giving for

Resulting in an index ellipsoid of

6x2 + y2

1

n2

+ p12Ssin(tKz) + z2

1

n2

+ p11Ssin(tKz) = 1


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with new principle indices of refraction of

nx = ny = n1

2n3p12Ssin(tKz)

nz = n1

2n3p11Ssin(tKz)

7

x2 + y2

1n2

+ p12Ssin(tKz)

+ z2

1

n2+ p11Ssin(tKz)

= 1


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ch 9.

Consider a transverse (shear) wave in GaAspolarized in the y direction, traveling in the z-direction

producing a strain wave of! ! ! which is

With the strain optic tensor for a cubic crystal

we get

u (z, t) = Ay cos(t kz)

Sij =ui

xj

Pij =

p11 p12 p120 0 0

p12 p11 p12 0 0 0

p12 p12 p11 0 0 0

0 0 0 p44 0 0

0 0 0 0 p44 0

0 0 0 0 0 p44

8

S4 = KA sin (t kz) = Ssin (tKz)

23 = 32 = p44Ssin (t

Kz)


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The modified index ellipsoid is

We can choose a new set of principle coordinates

x, y and z that recast this equation into theform

From inspection x=x, so let z,y and z,y differ bya rotation of around the x-axis:

z=zcos-ysiny=zsin+ycos

x2

n2

x

+y

2

n2

y

+z

2

n2

z

= 1

9

1

n2

x2 + y2 + z2

+ 2yzp44Ssin(t kz) = 1

(z cos y sin )2

n2

+(z sin + y cos )2

n2

+x2

n2

+ 2(z cos y sin )(z sin + y cos )p44Ssin(tKz) = 1


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Evaluating

gives

cross term 2zycos(2)0 for =45 giving

or

z

z

y y

x2

n2+

1

n2 Ssin(tKz)

y2 +

1

n2+ Ssin(tKz)

z2 = 1

10

(z cos y sin )2

n2+

(z sin + y cos )2

n2+

x2

n2+ 2(z cos y sin )(z sin + y cos )p44Ssin(tKz) = 1

x2

n2+

y2

n2+

z2

n2

zy sin2

n2+

zy sin2

n2+ (z2 sin2 + 2zy cos2 y2 sin2)p44Ssin(tKz) = 1

z

n2+ y

n2+ z

n2+ (z2

y2)Ssin(t

Kz) = 1


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Equating

to

gives

x

n2

x

+y

n2

y

+z

n2

z

= 1

so

x2

n2+

1

n2 Ssin(tKz)

y2 +

1

n2+ Ssin(tKz)

z2 = 1

n2z = n2

11 + n2Ssin(tKz)

1

n2+ Ssin(tKz)

=

1

n2z

n2

z n

2(1 n2Ssin(tKz))

n

z n1

1

2

n2Ssin(tKz) n

y n1 +

1

2

n2Ssin(tKz)nx = n

11


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ch 9.

Light will diffract from the periodic mediumproduced by the acousto-optic effect. Since theacoutic wave is slow compared to the light wave

(vs/c10-5), the acousto-optic perturbation can betreated as a stationary volume grating

Strong coupling to the diffracted wave occurswhen the Bragg condition is met:

for typical parameters this occurs for 5

2k sin = K

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The Bragg condition is equivalentto! ! ! ! ! found fromgeometrical consideration

The effect of motion of theacoustic wave is to Dopplershift the frequency of thediffracted wave according to

where vs sin is the component of the acoustic wavevelocity in the direction of the incident beam. At the Braggcondition! ! ! ! ! ! for the diffracted beam

Figure 9.2

2 sin = /n

= 2vs sin

c/n

=2vs

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ch 9.

Alternatively consider the opticalwave as made up of photons ofenergy and momentum k

and the acoustic wave as made upphonons of energy and momentum

K. For one photon absorbing one phonon

Conservation of energy requires =+

Conservation of momentum requires k=k+K

k,k,

K,

k

kk

K2

2k sin = K

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Picture of momentum conservationfor photons and phonons involved indiffraction in isotropic material is an

isosceles triangle since kk=n/c

In anisotropic material kk since n isa function of propagation angle nn.We can use the intersection of the

normal shells with the plane ofdiffraction to reconsider momentumdiagram. The Bragg condition canbe generalized to

k

k

kK

2

Propagation in thex-z plane of a uniaxial crystal,

acoustic wave at 30 from z axisoptical wave polarized as e-wave

k

Kk

sin + ksin

= K

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k

Kk

incident wave polarized as o-wave,diffracted wave polarized as e-wave

kK

k

k

Kk

kK

k

incident wave polarized as e-wave,diffracted wave polarized as e-wave

incident wave polarized as e-wave,diffracted wave polarized as o-wave

incident wave polarized as o-wave,diffracted wave polarized as o-wave

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For a given direction of the acoustic wave, there isa maximum value for K that allows Braggdiffraction. The maximum value can easily be found

from the geometry of the normal shells diagram

diffraction in the x-z plane of auniaxial crystal with an acoustic

wave propagating in the z-directionand incident wave o-polarized,diffracted wave e-polarized

diffraction in the x-z plane of auniaxial crystal with an acoustic

wave propagating in the x-directionand incident wave o-polarized,diffracted wave e-polarized

17Kmax = k0(no + ne)Kmax

= 2k0

n0


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diffraction in the x-z plane of abiiaxial crystal with an acoustic wave

propagating in the z-direction,

incident wave polarized in xz plane,diffracted beam polarized along y

diffraction in the x-z plane of abiaxial crystal with an acoustic wave

propagating in the z-direction,incident wave polarized in xz plane,

diffracted beam polarized in xz plane

Kmax = 2nxKmax = nx + ny

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ch 9.

Geometrical arguments give a constraint on theBragg angle for acousto-optic modulation, butdetermining the efficiency of conversion requires

we analyze the coupling of the undiffracted anddiffracted modes

Consider a total electric field

where A1 and A2 are coupled by the effect ofthe acoustic wave

E= A1E1e

i(1tk1.r) +A2E2e

i(2tk2.r)

ij = 0

ij= pijklSklcos (tKz)

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ch 9.

The perturbation to the impermeability tensor

is equivalent to a perturbation to the dielectrictensor

where 1 is a tensor of the first Fouriercoefficients of the tensor (z) and is given by

This perturbation will couple the fieldamplitudes A1 and A2

ij = 0

ij= pijklSklcos (tKz)

(z, t) = 21 cos(tKz) cos(tKz)

1 = (pijklSkl)

20

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ch 9.

The wave equation in a perturbed system can bewritten as

using the solutions in the unperturbed material

along with k2=2 and assuming A is slowly

varying (dA/dz


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ch 9.

Consider the cases of smallangle diffraction, shown in figure

9.7a of the textbookThe interaction region is thewidth of the acoustic beam andthe mode amplitudes are

functions of only x giving

k1

k2

2

1

12

22

2i1dA1

dxE1e

i(1t1x1z) 2i2

dA2

dxE2e

i(2t2x2z)

= 2

1e

i(tKz)+ 1e

i(tKz)

A1

E1ei(1t1x1z) +A2

E2ei(2t2x2z)


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k1

k2

2

1

12

The term is related to the asymmetry ofthe incident and diffracted waves relative tothe acoustic wave.

The conversion from A1 to A2 occurs when A1and A2 are in phase, but when they drift outof phase due to the acoustic wave converts field A2into field A1

When the Bragg condition is satisfied 1=2 so =0 and

the conversion efficiency is maximized

dA1

dx= i12A2e

ix,

dA2

dx= i

12A1e

ix,

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ch 9.

k1

k2

2

1

12

At the Bragg condition

which has solutions

where!! ! ! ! If A2(0)=0 this gives

dA1

dx= i12A2,

dA2dx

=i12A1

b = sin1

K

2k

= sin

1

2

A1 (x) = A1 (0) cos x i12

A2 (0) sin x,

A2 (x) = A2 (0) cos x i

12

A1 (0) sinx,

= |12|.

A1 (x) = A1 (0) cosx

A2 (x) = i

12

A1 (0) sinx 25


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The fraction of the power transfered to the diffractedbeam after an interaction distance x is

with full power transfer occurring at x=/2. Where can be expressed in terms of the original acousto-strainpijklSkl matrix

which depends on the intensity of the acoustic wave (viathe strain tensor Skl), the material properties, and thegeometry.

Idiffracted

Iincident =| A2 (x) |

2

| A1 (0) |2= sin

2

x,

=n3

4c cosB

|p1. (pijklSkl) p2|.

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ch 9.

Consider the cases of largeangle diffraction, shown in figure9.7b of the textbook

The mode amplitudes only varyalong z giving

giving k1

k2

2

1

1

2

E=A1 (z) E1e

i(1t1z) +A2 (z) E2ei(2t2z)

eix

dA1

dz= i

1

|2|12A2e

iz

dA2

dz= i

2

|2|12A1e

iz

=

1

2 Kwith where12 =

2

|12|

p

1 1p2 27


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For the case where 12>0 (i.e. forward diffraction)the solutions to

subject to the boundary conditions A2(0)=0 is

dA1

dz

= i12A2eiz

dA2

dz= i

12A1e

iz

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A1 (z) = A1 (0) ei1

2z

cos szi

2ssin sz

A2 (z) = iA1 (0) ei

1

2z

12

ssin sz

with s2 = |12|2 +

1

2

2


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The transfer of energy from A1 to A2 after interactionthrough a length L is given by

which is maximized when =0 (no phase mismatch ofincident and diffracted beams) and requires =0 ,L=/212 for 100% conversion.

=

|A2 (L) |

|A1 (0) |2 =

|12|

|12|2 +1

2

2 sin

2

sL.

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For the case where 12


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The transfer of energy from A1 to A2 after interactionthrough a length L is given by

which is identical to that of a Bragg reflector. Thisconfiguration can be thought of as a periodic Braggreflector, since the speed of the acoustic wave is somuch smaller than the speed of light that it can betreated as equivalent to a static perturbation of thepermittivity tensor.

R =

A2 (0)

A1 (0)

2

=

|12|2 sinh2 sL

s2 cosh2 sL+1

2

2sinh2 sL

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ch 9.

thus! ! ! ! ! giving

from which the index ellipsoid becomes

i = pi6S6

1

n2o

x2 +

1

n2o

y2 +

1

n2e

z2 2

ip41Ku0e

i(tKy)xz

i (p11 p1

1

n2o

x2 +

1

n2o

y2 +

1

n2e

z2 2

ip41Ku0e

i(tKy)xz

i (p11 p12)Ku0e

i(tKy)xy = 1

33


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sn.

Our expressions for the optical field diffracted fromthe acoustic wave assume there is only one diffractedorder (m=1). Is it possible to have|m|>1, corresponding to more than 1 phonon absorbed?

Conservation of energy requires k and k be almost thesame length (except for the relatively small increase ofk due to absorbed energy of phonon). This isntpossible for more than one order at a time, with a

unique K vector.

34

k

kk

K2k

k1

K2

k2


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sn.

If the acoustic wave is a finite beam with width X,Heisenbergs uncertainty relation xp/2 tells us it

will have a range of K-vectors K=1/(2X).

If the range of K-vectors is large compared to theBragg angle, multiple phonos to be absorbed withoutchan in len th of k relative to k.

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k

kk

K2k

k1

K2

k2

k

K2

k1k

2


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sn.

Acoustic beam spread is K=K

Thus the Bragg angle (b=/(2n) )can beexpressed in terms of the acoustic beam spread

Q>1 is called the Bragg regime and correspondsto single order diffraction

Q


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sn.

Consider a narrow acoustic beam introducing achange in the index of refraction in a material of

an optical wave expressed by

is incident at x=0, upon exiting the acoutic beamat x=X it can be written as

with

where is the direciton of the optical beam

relative to the x-axis37

E= E0ei(tkr)

n(0 < x < X) = n0 sin(t K r)

E= E0ei(tkr)

=

X

0

1

cos

cndx


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sn.

IfX is small (i.e. in the Rama-Nath regime), index n(x,t) seen by optical beam can be considered constantacross beam width

giving

where

is called the modulation index. This can be expandedusing a form of the Jacobi-Anger identity

38

=X

cos cn =

X

cos cn0 sin(t

K

r)

E= E0ei(tkr sin(t Kr))

X

cos

cn0

ei sinx =

m=

Jm()eim


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sn.

Expanding

into Bessel functions gives

and the diffraction efficiency for the mth orderdiffracted beam is

39

E= E0ei(tkr sin(t Kr))

E= E0

m=

Jm()ei((m)t(km K)r)

m = J

2

m() = J2

m

X

cos

cn0


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sn.40

m = J2

m() = J2

m

X

cos

cn0

1 2 3 4 5 6 7 8 9 1

-

-0.5

0.5

J0()

J1() J2()


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sn.

Calculate the acoustic beam width X tomaximize diffraction into the m=1 order forn0=0.0001, =00 and =633 nm. What fraction

of the power gets diffracted into the m=1 beam?J1() is max at 1.85, solving for X we getX=1.9 mmEvaluating =J12(1.85)0.33

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ch 9.

Yariv & Yeh Optical Waves in Crystals chapter 9

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Ch 9-Acoustooptic Effect

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