Chapter 12. Interaction of Light and Sound
description
Transcript of Chapter 12. Interaction of Light and Sound
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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
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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
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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
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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
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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
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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
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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
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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
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])[(
])[(
)(
]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
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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
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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
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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
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])([00
2)(
])([])([0
2)(00
2
12
2
12
zkxizxi
zkxizkxizxi
eAi
ez
A
x
Ai
eAeAi
ez
A
x
Ai
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(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
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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
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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
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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
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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
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(2) Large Bragg angle diffraction
02/
x
A
])([
002)(
])([])([0
2)(0
4
14
1
zKxizxi
zKxizKxizxi
eAez
A
eAeAez
A
x-dependent term
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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
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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
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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 :
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0)(
~,1)0(
~0 DzAzA
gDgDg
gDgADP
gDgDg
gADP
222122
222122
222122
22
00
sinhcosh
sinh~)(
sinhcosh
~)(
# Application : DBR reflector
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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
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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,
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,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
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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
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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