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
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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 moleculesNonlinear Optics Lab . Hanyang Univ.
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.
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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)
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Substituting from (19.1-2) into (19.1-5), Is
(19.1-6)
where, : Figure of merit for the strength of the AO effect in the material
Consequently, refractive index of medium is
where,
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Nonlinear Optics Lab . Hanyang Univ.
Nonlinear Optics Lab . Hanyang Univ.
Nonlinear Optics Lab . Hanyang Univ.
Figure of merit of the material relative to water
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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
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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
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] [
)
sin(qx
ej(qx) ej(qx) 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.
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Bragg condition (upshifted reflection)
Sinc function has maximum when the argument is zero: 2k sin
q
q 0(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.
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Bragg condition (𝑞 = 2𝑘𝑥 sin 𝜃𝐵) is equivalent to the vector relation,
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Tolerance in the Bragg condition
(19.1-14)
] )2
sin sinc[(sin
] 2 )
sin (sin
2 sinc[
] 2 )
sin 2 sinc[(
q
q
p q
q p
q
/ L
/ L k
/ L q k
B B
q k sin
q
B 2First zero at .
Since L is typically much greater than . This is an extremely small angular width.
L
B / 2
sin
sin
q
q
q
q
B
/ 2LNonlinear Optics Lab . Hanyang Univ.
Doppler shift
(19.1-14) r exp( jt)
Incident light field is proportional to [ ], so the reflected wave has the form of , and has angular frequency
) exp( j t
E
) exp( j t
] ) (
exp[ j t E
r
E
r
: 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
vs r
vssinq
Doppler shifted frequency :
L
L
r [1 2( / 2p)( / 2 )(2p / )]
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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)
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From (19.1-18), by using ,
or, by (19.1-6)
sinq=/2L
- R : : typical light-
04
4 scattering phenomena.- R : valid only low intensity region since this result is obtained from the approximation theory (19.1B : Coupled-wave theory)
I
s
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Downshifted Bragg Diffraction
q k sin
q
B 2Nonlinear Optics Lab . Hanyang Univ.
Quantum interpretation
)
(
,k (
r ,k)) (,q
Photon (, k) interacts with acoustic phonon (, q), and generates a new reflected photon (r , k)
Energy and momentum conservation conditions:
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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
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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),
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Coupled-wave equations (upshifted Bragg diffraction)
where,
(19.1-28)
Refer to Appendix II ~…
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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 : Ar(0) 0
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Reflectance :
D
0 2 0
sin
p n d
2
0 0
D
p n d
d
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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: 1𝐷𝜆, Circular: 1.22𝐷𝜆, Gaussian(waist diameter=2w0 ): 𝜋𝑤𝜆
0 = 0.64𝐷𝜆
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.
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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.
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Diffraction of an optical plane wave from a thin acoustic beam :
Raman-Nath Diffraction
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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
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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 2 0
sin
p n d Re
(19.1.B), Reflectance, can be unity : 100% reflection
Efficient frequency shifter
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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!
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Bandwidth :
or : Transit time of sound waveNecessary 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.
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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~
Nonlinear Optics Lab . Hanyang Univ.
[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~
Nonlinear Optics Lab . Hanyang Univ.
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~
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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
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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
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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.
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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.
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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.
Nonlinear Optics Lab . Hanyang Univ.
Nonlinear Optics Lab . Hanyang Univ.
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.
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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
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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 .
33 23
32 13
31
23 22
12 21
13 12
11
h h
h h
h
h h
h h
h h
h
Symmetrical tensor (example):
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Example) Strain-optic matrix for cubic crystal (or isotropic media):
Additional constraint: There are only two independent coefficients!
(miss print !)
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0 0 0
) cos(
) cos(
) cos(
0 0 0
) cos(
0 0
0 0 0 0 0
0 0
0 0 0
0 0 0
0 0
0 0 0
0 0 0
0 0 0
0 11
0 12
0 12
0
44 44 44 11 12 12
12 11 12
12 12 11
qz t S
p
qz t S
p
qz t S
p
qz t S
p p p p p p
p p p
p p p
s pIK ij
From example 19.3-1,
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z
Nonlinear Optics Lab . Hanyang Univ.
Bragg diffraction (Refer to Appendix II~)
Bragg condition:
(z-component)
(x-component)
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From (19.3-18), when 𝜃𝑟 = 𝜋/2
L
sin 0
cos 0
q q n n
n
r
2 p / q
2 )
1 q p /
L
0 n nr 2
)
2 q p /
L
0 n
nr
Nonlinear Optics Lab . Hanyang Univ.
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
Nonlinear Optics Lab . Hanyang Univ.
Photoelastic effect
: Mechanical strain Index of refraction
Index ellipsoid for Principal axes : 2 1
3 2 2
2 2 2
1 2
n
z n
y n
x
Strain tensor elements, S :
zxxz
zy yz
yx xy
zz yy
xx
x S z w
S u
y S z w
S v
x S y v
S u
w z y S
S v u x
S
, ,
where, u, v, w : displacements along the x, y, z axes
xy zx
yz
zz yy
xx
S S
S S
S S
S S
S S
S S
6 5
4
3 2
1
, ,
,
, : Tensile(Normal) strain
: Shear strain
I. Photoelastic effect
Nonlinear Optics Lab . Hanyang Univ.
Change in index of refraction due to the mechanical strain :
) 6 , , 1 ( 1 )
(
6
1
2
D
i S n i j pij j
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 :
1 2
2 2
1 1
1
6 5
4
2 3 3 2 2 2
2 2 2 1
1 2
j
j j j
j j j
j j
j
j j j
j j j
j j
S p xy S
p xz S
p yz
S n p
z S
n p y S
n p x
Yariv, P. Yeh,
“Optical Waves in Crystals”, John Wiley & Sons
Nonlinear Optics Lab . Hanyang Univ.
Nonlinear Optics Lab . Hanyang Univ.
Nonlinear Optics Lab . Hanyang Univ.
Example) Sound wave propagating along the z direction in water
Sound wave : w(z,t)Azˆcos(tKz)
0 ) / ( ) / ( ,
0 ) / ( ) / ( ,
0 ) / ( ) / (
) sin(
) sin(
/ ,
0 /
, 0 /
6 5
4
3 2
1
x v y
u S
S x
w z
u S
S y
w z
v S
S
Kz t S
Kz t KA
z w S
S y
v S
S x
u S
S
xy zx
yz
zz yy
xx
Elasto-Optic Coefficient for the water (isotropic, Table 9.1) :
) 2(
0 1 0
0 0 0
0 )
2( 0 1
0 0 0
0 0
) 2(
0 1 0 0
0 0
0
0 0
0
0 0
0
12 11 12
11 12
11 11
12 12
12 11 12
12 12 11
p p p
p p
p p
p p
p p p
p p p
pij
0 1 )
(
), sin(
1 ) (
), sin(
1 ) ( 1 )
(
6 , 5 , 2 4
11 2 3
12 2 2
2 1
D
D
D
D
n
Kz t S
n p
Kz t S
n p n
The new index ellipsoid :
1 ) 1 sin(
) 1 sin(
) 1 sin(
2 11 2
2 12 2
2 12
2
p S t Kz
z n Kz
t S
n p y Kz
t S
n p x
Nonlinear Optics Lab . Hanyang Univ.
Example) y-polarized Shear wave propagating along the z direction in Ge
Sound wave : v(z,t)Ayˆcos(tKz)
0 ) / ( ) / ( ,
0 ) / ( ) / (
), sin(
) sin(
) / ( ) / (
, 0 /
, 0 /
, 0 /
6 5
4
3 2
1
x v y
u S
S x
w z
u S
S
Kz t S
Kz t KA
y w z
v S
S
z w S
S y
v S
S x
u S
S
xy zx
yz
zz yy
xx
Elasto-Optic Coefficient for the Ge (cubic, Table 9.1) :
44 44 44 11 12 12
12 11 12
12 12 11
0 0 0 0 0
0 0
0 0 0
0 0 0
0 0
0 0 0
0 0 0
0 0 0
p p p p p p
p p p
p p p
pij
0 1 )
(
), sin(
1 ) (
, 0 1 )
( 1 )
( 1 )
(
6 , 2 5
44 2 4
2 3 2 2
2 1
D
D
D
D
D
n
Kz t S
n p
n n
n
The new index ellipsoid :
1 ) sin(
2 ) 1 (
44 2
2 2
2 x y z yzp S tKz
n
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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 ;
sin( )
ee 2
2 0
2 t Kz
t D
2 2 2 0
2 0
2 ) sin( )
( t
Kz e t
t e
D
2 2 ) ( )
( 2 0
2 2 0
2 2
2
) 2 (
1
t e e
i e t
e z
e x
e i t Kz i t Kz
D
: ) dependence y
(no problem D
- 2 to converting
e e e e 0
: e-field Total
where,
] )
[ )
(
) (
0 ) ( 0
0
) , ( )
, (
) , ( )
, (
z x t i t
i
z x t i t
i
e z x A e
z x A e
e z x A e
z x A e
r k
r k
Nonlinear Optics Lab . Hanyang Univ.
2 2 2 0 2
2 2 2 0 2
)
(
k k
z k A z
A x
k A x
A
2 2 2
2
,
Let,
And, slow varying approximation ;
i[( t x z)] i[( )t x z)] i[( )t x z)]
) Kz t ( i )
Kz t ( i
)]
z x t ) [(
i
)]
z x t ) [(
i
) z x t ( i
e A ) (
e A ) (
e A
e i e
z e A x
i A
z e A x
i A
z e A x
i A
D
2 2
0 0
0 0
2 1 2 2 2
Nonlinear Optics Lab . Hanyang Univ.
] ) ( [ 0 0
) 2 (
] ) (
[ ]
) (
[ 0
2 )
0 ( 0
2 2 1
2 2 1
z K x
z i x i
z K x
i z
K x
z i x i
e i A
z e A x
i A
e A e
i A z e
A x
i A
D
D
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(1) Small Bragg angle diffraction
0
0
z
q A
] ) ( [ 0 0
2 )
(
] ) (
[ ]
) (
[ 0
2 )
0 (
2 2 1
2 2 1
z K x
i z
x i
z K x
i z
K x
i z
x i
e i A
dx e i dA
e A e
i A dx e
i dA
D
D
K
K
& : Bragg condition
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D
D
D
D
A A
A A
e A dx k
A d
e A dx k
A d
ix ix
2 / 1
0 0 2 / 1
0 0
2 / 0 1
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 0 2
2 0
2 D
A
dx A i d
dx A
d
4 2
1/22 1
) (
0 ) (
0 0
) ( )
~ (
: solution
where,
2 1 2
1
D
D D
k
e D e
C x
A ix ix
2
0 12 0 1
) (
) (
, )
~ (
where
2 1 2
1
i D D
i C C
e e
D e
C x
A ix ix ix
D
D
D
D
D
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Diffraction efficiency, h
D
D
4 1 2 1 4 0
1 2 1 0 0
,
condition) (initial
0 ) 0
~ ( , 1 ) 0
~ (
D C
x A x
A
sin ( ))
~ ( ) (
) ( 2 sin
) ( cos )
~ ( ) (
2 2 2
2 2 2 2
0 0
x x
A x P
x x
x A x P
D
0-th and 1-st diffraction powers :
i) P0(x)P(x)1
2214(D)2
ii) Maximum transfer : D0; )
( sin ) (
) ( cos ) (
2 2 0
x x
P
x x
P
Diffration efficiency :
2 , ,3 1 2
) ( sin )
( 2
p p
h
h
L
L L
p
