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Acousto-optic modulators and deflectors
An RF signal is applied to an impedance matching network that is connected across theelectrodes of a thin piezoelectric transducer that is bonded to a photoelastic material(oriented, cut, polished crystal or isotropic glass) which launches an acoustic wave intothe interaction medium. The acoustic wave induces a traveling-wave volume dielectricgrating pertubation through the photoelastic effect which can efficiently diffract anappropriately aligned optical wave (Bragg matched and and polarization eigenmode)producing a modulated output.
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 1
Historical overview of Acousto-optic devices
• 1922 - Brillouin predicted the light diffraction by an acoustic wave
• 1932 - Debye and Sears, Lucas and Biquard carried out first AO experiments
• 1937 - Raman and Nath analyzed AO interaction taking into account several orders(Raman-Nath regime)
• 1956 - Phariseau developed this model including Bragg diffraction
• The invention of laser, crystal growth technology and high frequency piezoelectrictransducers, has led to the development of acousto-optics technology
•Modulators, Deflectors, Q-switches, frequency shifters and Tunable filters
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 2
Acousto-optic Interactions
∆ǫ = ε0W
2ǫpSǫei(Ωt−
~kn·~x) + cc
~Em(~x, t) = emAm(z)ei(ωmt−~km·~x) + cc
Conservation of Energy
photon phononE = ~ωm E = ~Ω
~ω0 ± ~Ω = ~ω1
ω0, ω1 ∼ 2π.5× 1015 rad/secΩ ∼ 2π109 rad/sec =⇒ λ0 ≈ λ1
Conservation of Momentum
photon phonon
~pm = h~km ~pA = h~KA
~k1 = ~k0 ± ~KA
|~km| = 2πλm/n =⇒ |~k0| ≈ |~k1|
k1
k0
KA
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 3
Bragg condition
sin θB =λ/2
ΛBragg Angle
θB
θB
KG
kd
ki
2πnλ
Λ
θB
λ/2 λ/2
ΛθB
Λ =vaf
Grating spacing =velocity
frequency
θB = sin−1|~K|2|~k|
= sin−1λ
2nΛ= sin−1
λf
2van≈ λf
2van
Angle ∝ frequency
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 4
Doppler shift
∆ω
ω=
2v
c
Velocity v
c
c
τ=λ/2c
c/nv
θv
v
In a medium with index n and with v not parallel to c/n v‖ = v sin θ
∆ω =2ωv‖c/n
=2ωv sin θ
c/n= 2
2π
λ/nv sin θ
using Bragg’s relationsin θ =
λ/n
2Λ=
K
2k
∆ω = 22π
λ/nvλ/n
2Λ=
2πv
Λ= Ω
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 5
Acousto-optic devices
AO mediumPiezo-electric transducer
Patterned Top electrode
Bottom electrode
Matching + Bonding Layers
v(t)
d
S(t)
BraggAngle
DC Beam
+1 Diffracted Beams
AOD
Piezoelectrictransducer
B
fhigh
flow
fhigh
flow
∆ǫij(~r, t) = −U0
2ǫikpklmnSmnǫlj cos(Ωt− ~K · ~r)
~E(~r, t) = Ai(z)pie−i(ωit−kxix−kziz)+Ad(z)pde
−i(ωdt+kxdx−kzdz)
(∇2 + ω2µ(ǫ +∆ǫ(~r, t))) ~E(~r, t) = 0
Index perturbation by acoustic wave(elasto-optic effect)
Electric-field in interaction region
inhomogeneous wave eq with dielectric perturbation
dAi
dz= −iκAie
i∆Kz
dAd
dz= −iκAde
i∆Kz
Coupled-mode equation
Ai(z) = Ai(0) cos |κ|z
Ad(z) = −iκ
|κ|Ai(0) sin |κ|z
η =|Ad(L)|2|Ai(0)|2
= sin2π2PsL
2λ2HM2
1/2
M2 =n6p2effρV 3
a
peff = p∗dǫpSǫpi
RF impedance matching
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 6
Acousto-optics is in same geometry as volumeholograms
λ/2
Λ
θB
KG
kd
ik
2πn/λ
Recording a volume hologram
L/L =40
Bragg-matchedvolume hologram
diffraction
θBλ
θB
ΛθB
0L
sin θ =λ/2 Λ
B sin θ =K /2 k
B G
sin θB = λ/2Λ
sin θB ≈ θB = KG/2k
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 7
Coupled Mode Equations
Coupled Mode Theory for Acousto-optic Volume Diffraction
∂A1
∂z= iκA2e
i∆kzeiΩt
∂A2
∂z= iκ∗A1e
−i∆kze−iΩt
At the Bragg angle θB = sin−1(
λ2Λ
)
I.C. A2(0) = 0, soln
A1(x) = A1(0) cos |κ|zA2(x) = −i κ
∗
|κ|A1(0) sin |κ|ze−iΩt
DE η = 100% at κL = π is limited by otherdiffraction orders not included in this simple2-mode coupled mode theory to 90-97%
We will see other orders in the lab
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 8
Bragg Mismatched Readout
∂A1
∂z= −iκA2e
i∆kzeiΩt
∂A2
∂z= −iκ∗A1e
−i∆kze−iΩt
constant of system (by conservation of energy)
∂
∂z(|A1|2 + |A2|2) = 0
When A2(0) = 0solution
A1(z) = ei∆kz/2
[cos sz − i∆k
2ssin sz
]A1(0)
A2(z) = e−i∆kz/2
[−iκ
∗
ssin sz
]A1(0)e
−iΩt
s2 = κ∗κ +(∆β2
)2
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 9
Intensity Diffration Efficiency
I2(L)
I1(0)=
|κ|2
|κ|2 +(∆k2
)2 sin2√|κ|2 +
(∆k
2
)2
L
for small ∆kI2(z)
I1(0)≈ sin2 |κ|z
for small κI2(z)
I1(0)≈ |κ|2snc2
(∆kz
2
)= |κ|2sinc2
(∆kz
2π
)
∆KL = 5 ∆KL = 10 .
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 10
Momentum space description of Acousto-opticDiffraction
K→
Induced polarization
propagating waves
K→
P ∝ E e A eik·r iK·r
→→ → →
|k|=2 π /λ→
momentum uncertainty
L
W
∆k=2 π/L
∆k=2 π/W
=EA ei(k+K)·r
→→ →
k→
K→
+
k→
Incident wave
L
Acoustic transducer
Diffracted wave
AOmedium
A
H
W
x
z
ki
kd
KA
k∆z
θ
Opticalmomemtumsurfacek
2πA
2πL
kz
kx kx
η
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 11
Acousto-optic Volume gratings, ~k-space and~K-space
g(~r) = [1 +m cos( ~KG · ~r)]Π( x
X
)Π( yY
)Π( zZ
)∗ ∗ ∗ h(~r)
=
∫ ∫ ∫G(~k)H(~k)ei
~k·~rd3~k
where h(~r) is the acoustic impulse response (transducer size and BW limited)
while H(~k) is the transducer bandwidth and size limited 3-D frequency response
G(~k)=
[δ(~k)+
1
2δ(~k− ~KG)+
1
2δ(~k+ ~KG)
]∗∗∗[Xsinc(kxX)] [Y sinc(kyY )] [Zsinc(kzZ)]
optical wavevectoror k-space
KGko
k r
KG
-KG
Grating space or K -spaceG
2πn/λ
yk
xk
kz
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 12
Grating Uncertainty
Finiteness of transducer and crystal size leads to a distribution of grating vectors dueto Fourier uncertainty
Bragg matchingAngular selectivityWavelength selectivity
k2
k1
ko01K
12K02K
Grating writing
kr
Diffraction in k-space
2π/Z
2π/X
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 13
Angular Selectivity
θ z
Recording
∆kz
∆θ
Bragg Matched Readout
sθ
δk z
To first order momentum surface is a flatsurface tilted by ±θ on both input and out-put beams. Rotating input beam away fromexact Bragg matching angle by ∆θ will givea z component to the motion of the k vec-tors given by
sin θ =δkzs
where s =2πn
λ∆θ =⇒ δkz =
2πn
λ∆θ sin θ
vector sum of ~ki + ~KG shifts away from the output surface by δkz too, giving
∆kz = 2δkz = 22πn
λ∆θ sin θ
The intensity DE is given by power of sinc function evaluated on momentum surface
η2(∆θ) = |φ|2sinc2(
∆kzL/2π
)= |φ|2sinc2
(2n sin θ∆θL
λ
)
with the first zero of the sinc null at θ0 =λ
2nL sin θand φ = πn1L
λ cos θ
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 14
Wavelength selectivity of acousto-opticgratings
θ z
Recording
∆k =2ptan θz
Wavelength Shifted Readout
2π/λ2πn/λ
θ p=sin θ (k -k )g r
θ
θ
Alignment for maximum DE at λ1 then read-out at shifted wavelength λ2. Produces ~koffset p = sin θ(|~k2| − |~k1|).Grating vector of length KG + 2p would be Bragg matched.Leads to a phase mismatch in z
∆kz = 2p tan θ = 2 tan θ sin θ(|~k2| − |~k1|) = 2sin2 θ
cos θ
(2π
λ2− 2π
λ1
)
Can simply realign for Bragg matching at the new wavelength.
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 15
Momentum space (~k-space and ~K-space)
Coupled wave equation with a dielectric perturbation δǫ(~r, t)
∇2[Ei(~r, t) + Ed(~r, t)] +1
c2∂2
∂t2[ǫr + δǫ(~r, t)][Ei(~r, t) + Ed(~r, t)] = 0
Separate into the coupled equations for the incident field of frequency ω and eachmonochromatic angular frequency component, ωm, of the diffracted field by includ-ing the harmonic temporal dependence and then using the othogonality of differentfrequency components to enforce the conservation of energy.
∇2Ei(~r, ω) +ω2ǫrc2
Ei(~r, ω) = −ω2
c2=∑
m
δǫ∗(~r,Ωm)Ed(~r, ωm)
∇2Ed(~r, ωm) +ωm
2ǫrc2
Ed(~r, ωm) = −ωm
2
c2δǫ(~r,Ωm)Ei(~r, ω)
Conservation of energy gives ωm = ω + Ωm
δǫ(~r,Ωm) is component of the dielectric perturbation oscillating at frequency Ωm.
Simplify using ω ≈ ωm since Ωm ≪ ω.
Solutions for different frequency components of Ed(~r, ωm) obey linear superposition
Born approximation: incident wave Ei is much stronger than the diffracted wave Ed.
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 16
Momentum space derivation continued
Represent a particular frequency component of the diffracted wave as a transverseFourier expansion of its plane wave components in the unperturbed media, whichevolves along the nominal direction of propagation z
Ed(~r, ωm) =
∫Eωmd (~kt, z)e
i~kt·~reikzd(~kt)zd~kt
kzd(~kt) =√k2d − ~kt · ~kt is the longitudinal component of the wavevector,
~kt = xktx + ykty is the transverse component of the wavevector,
kd = 2πnd(~kt, ωm)ωm/c is the magnitude of the diffracted wavevector
nd(~kt, ωm) accounts for material anisotropy and dispersion (negligible for AOD)
Substitute spectral decomposition:∫ [2ikzd(~kt)
∂
∂zEωmd (~kt, z) +
∂2
∂z2Eωmd (~kt, z)
]ei~kt·~reikzd(
~kt)zd~kt = −ω2m
c2δǫ(~r,Ωm)Ei(~r, ω)
SVEA neglects second derivative kzd(~kt)∂∂zEωmd (~kt, z)≫ ∂2
∂z2Eωmd (~kt, z)
∫ ∫2ikzd(~kt)e
ikzd(~kt)z
∂
∂zEωmd (~kt, z)e
i(xktx+ykty)dktxdkty = −ω2m
c2δǫ(~r,Ωm)Ei(~r, ω)
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 17
Momentum space derivation continued
Take transverse Fourier transform in x and y
eikzd(kx,ky)z∂
∂zEωmd (kx, ky, z) =
iω2m
2c2kzd(kx, ky)
∫ ∫δǫ(~r,Ωm)Ei(~r, ω)e
−i(xktx+ykty)dxdy
Integrate directly to yield the field of the diffracted wave Ed at the exit face, z = L
Eωmd (kx, ky, L) =iω2
m
2c2kzd(kx, ky)
∫ z=L
z=0
FT xyδǫ(~r,Ωm)Ei(~r, ω)e−ikzd(kx,ky)zdz
Reformulated as a 3-D Fourier transform by noting that δǫ(~r,Ωm) vanishes outside theregion z ∈ 0, L.
Eωmd (kx, ky, L) =iω2
m
2c2kzd(kx, ky)
∫δ(kz − kzd(kx, ky))FT xyzδǫ(~r,Ωm)Ei(~r, ωm)dkz
This is key result. It states that the angular spectrum components of the diffracted fieldat the output of the media containing the weak dielectric perturbations is given by the 3-D Fourier transform of the product of the incident field and the dielectric perturbation,evaluated on the surface of the allowed propagating modes for the diffracted field.
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 18
Numerical Calculations of Momentum Space inBraggart
TeO2 k-space
-1•107 0 1•107
k (1/m) in [001]
-1•107
0
1•107
k (1
/m)
in [1
10]
TeO2 k-space
1.6820•1071.6822•1071.6824•1071.6826•1071.6828•107
k (1/m) in [001]
-2•105
-1•105
0
1•105
2•105
k (1
/m)
in [1
10]
TeO2 k-space
1.68235•1071.68240•1071.68245•1071.68250•107
k (1/m) in [001]
2.355•105
2.360•105
2.365•105
k (1
/m)
in [1
10]
L
A
λ
Λ
Acoustic beam of width L
Optic beamof widthA
k = 2 π∆ / L
k = 2 π∆ / A
a)
b) c)
16.82 16.8816.85k (10 /m) in [001]6
d)
16.8235 16.8250k (10 /m) in [001]6
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 19
Acousto-optic device in Momentum space(k-space)
Undiffracted beam
Acousto-optic Deflector
Λ
A
Diffracted beam
RF signalL
Acousto-optic Bandshapewith 3dB midband rippleDE
f0f1pmf
uflf
f2pm
B
2πΑ
2πL
k2
k1
Kx
Ky
KG= 2πΛ
g(~r) = [1 +m os( ~Kg)re t xAre t yYre t zL FT
G( ~K) = [Æ( ~K)+Æ( ~K ~Kg)+Æ( ~K+ ~Kg)AY Lsin (Akx) sin (Lkz)sin (Yky)• Real space = FT(k-space)Robert McLeod, “Spectral-Domain Analysis and Design of Thr ee-Dimensional Optical Switching and Computing Systems”, Phd Thesis, U. Colorado 1995
R. T. Weverka, K. Wagner, R. Mcleod, K. Wu, and C. Garvin, “Low-Loss Acousto-Optic Photonic Switch”, Acousto-optic signal processing, N. Berg and J. Pellegrino, Eds, 1996
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 20
Isotropic Bragg Cell
H
MatchingNetwork
GroundPlane
Piezoelectric Transducer Top
Electrode
Angle cutback facet
TransparentPhotoeleastic
Medium
L
Isotropic Bragg Cell
aK
ki
kd
fmax
f0
fmin
FT Ε(r)δε(r)=E(k)*δε(k)
L
A
Ε(r)δε(r)
Acousto-optic BandshapeDE
f0 fmaxfminf
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 21
Acousto-optic Diffraction Efficiency
At Bragg angle incidence, ∆k = 0
η = sin2(∆φ
2
)= sin2
(k0∆nL
2 cos θ
)
Definition of the photoelastic effect, δη = p S gives
∆n = −n3peffS
2where effective tensor element is peff = po ǫ p S ǫ pi
Acoustic energy density given material density ρ and acoustic velocity V is ρV 2S2
2
[Jm3
]
Thus the average energy flow (eg acoustic power) for a transducer of area A = HL is
Pa = V HLρV 2S2
2[W]
so the peak efficiency is
η = sin2
(π
λ0 cos θ
√M2L
2HPa
)≈(
π
λ0 cos θ
)2M2L
2HPa
Where the AO material figure of merit is M2 =n6p2
ρV 3
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 22
pepSep and perEep
the pertubation of the impermeability for EO and AO is
∆η = r ~E(0) ∆η = p S
But since we are used to the dielectric tensor in the wave eqn we need ∆ǫ
ǫ η=ǫoεrηr
ǫo=η ǫ =
ηr
ǫoǫoεr=I (η+∆η)(ǫ−∆ǫ)=η ǫ−η ∆ǫ+∆η ǫ−∆η ∆ǫ=I
Now for small pertubations we can neglect the last term, and cancel η ǫ = I
η ∆ǫ = ∆η ǫ
Multiply through by ǫ gives
∆ǫ = ǫ ∆η ǫ ∆εr = εr ∆ηr εr
Now, this gives a polarization source term
~P = P0pP = ǫo∆χ ~E = ǫo ∆εr ~E = ∆ǫ piE0
z z
y
x
o
S kPe de
Which can only radiate if it aligns with a momentum and energy matched propagatingmode po · pP . These geometric projections give an effective tensor coefficient
reff = p∗o εr r E(0) εr pi peff = p∗o εr p S εr pi
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 23
reff and peff
Note that
d =ǫp
|ǫp| =εrp
|εrp|Thus we can associate an effective EO and AO coefficient with the quadratic forms
found by left and right projecting the ~D unit eigenvector onto ∆η
reff = d∗o r E(0) di peff = d∗o p S di
On the other hand the scalar coupling coefficient needed in the coupled mode eqns is
κreff = p∗o εr r E(0) εr pi κp
eff = p∗o εr p S εr pi
which for a uniaxial crystal and a choice of ordinary or extraordinary modes is
κroo = o∗o εr r E(0) εr oi = n4
oreff κpoo = o∗o εr p
S εr oi = n4
opeff
κroe = o∗o εr r E(0) εr ei = n2
on2e(θi)reff κp
oe = o∗o εr pS εr ei = n2
on2e(θi)peff
κreo = e∗o εr r E(0) εr oi = n2
e(θo)n2oreff κp
eo = e∗o εr pS εr oi = n2
e(θo)n2opeff
κree = e∗o εr r E
(0) εr ei = n2e(θo)n
2e(θi)reff κp
ee = e∗o εr pS εr ei = n2
e(θo)n2e(θi)peff
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 24
Effective Photoelastic Constant:Example Cubic Crystals
Acoustic wave with paricle motion u, direction ~K yields unit Shear S = 12
[∂ui∂rj
+∂uj∂ri
]
Optical input polarization pi gives di =ǫpi|ǫpi| and output polarization po gives do =
ǫpo|ǫpo|
Then we can calculate the effective photoelastic constant as
peff = d∗opS di = d∗o
[p6×6 S1×6
]Unfold
di
or the coupling coefficient in the scalar coupled mode equations as
κ = p∗o εr p S εr pi
z
xθ θ’
θθ’
ki
kd
KA
sp
sps’^
p’^s’^
p’^
z
x
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 25
Photoelastic Coupling
p⇒ p ≡ pIJ =
p11 p12 p12 0 0 0p12 p11 p12 0 0 0p12 p12 p11 0 0 00 0 0 p44 0 00 0 0 0 p44 00 0 0 0 0 p44
Sij =
S11 S12 S13
S21 S22 S23
S31 S32 S33
=
S1
12S6
12S5
12S6 S2
12S4
12S5
12S4 S3
⇒ S ≡ SJ =
S1
S2
S3
S4
S5
S6
∆η = pS ⇒
∆η1∆η2∆η3∆η4∆η5∆η6
=
p11 p12 p12 0 0 0p12 p11 p12 0 0 0p12 p12 p11 0 0 00 0 0 p44 0 00 0 0 0 p44 00 0 0 0 0 p44
S1
S2
S3
S4
S5
S6
⇒
∆η1 ∆η6 ∆η5∆η6 ∆η2 ∆η4∆η5 ∆η4 ∆η3
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 26
Acoustic Waves
Cubic crystals are optically isotropic so θ = θ′~ki =
2πnλ (cos θ, 0,− sin θ) esi = (0, 1, 0) epi = (sin θ, 0, cos θ)
~kd =2πnλ (cos θ, 0, sin θ) esd = (0, 1, 0) epd = (− sin θ, 0, cos θ)
Acoustic wave along z is pure longitudinal or 2 shears ~KA = 2πΛ (0, 0, 1)
Longitudinal~ul(~r, t) = zW cos(Ωt− ~Kl
zz) Sl(~r, t) = zzWK lz cos(Ωt− ~Kl
zz)ˆS→ S33→ S3
x shear~us(~r, t) = xW cos(Ωt−~Ks
zz) Ss(~r, t) = xzWKsz cos(Ωt−~Ks
zz)ˆS→ S13→ S5
y shear~us(~r, t) = yW cos(Ωt−~Ks
zz) Ss(~r, t) = yzWKsz cos(Ωt−~Ks
zz)ˆS→ S23→ S4
Longitudinal Acoustics S3
s→ speff =
[0 1 0
]p12 . .. p12 .. . p11
010
= p12
s→ p
peff =[− sin θ 0 cos θ
]p12 . .. p12 .. . p11
010
= 0
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 27
Shear wave Diffraction
x Shear S5
s→ s
peff =[0 1 0
]
. . p44
. . .p44 . .
010
= 0
s→ p
peff =[− sin θ 0 cos θ
]
. . p44
. . .p44 . .
010
= 0
y Shear S4
s→ s
peff =[0 1 0
]. . .. . p44. p44 .
010
= 0
s→ p
peff =[− sin θ 0 cos θ
]. . .. . p44. p44 .
010
= p44 cos θ
Since θ increases with f this peff is frequency dependent
cos θ =√
1− sin2 θ =
√1−
(λf2nVa
)2peff = p44
√1−
(λf2nVa
)2
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 28
Bragg cell examples
Crystal GaP TeO2 LiNbO3 PbMO4
Mode L[110] S[110] L[100] L[001]Velocity 6.32 .62 6.57 3.63mm/µsecAttenuation 3.8 17.9 .1-1 5.5
dB/µs/GHz2
M2 =n6p2
ρv329.5 795 4.6 23.9
B 1GHz 50MHz .7GHzT .6µsec 50 µsec 4µsecTB 600 2500 2800Efficiency 8%/Watt 200 %/W 2%/W
Self Collimating Opt Activity Anisotropic IsotropicPol Switching splitting diffraction geometry
Cost $10KKelvin Wagner, University of Colorado Advanced Optics Lab 2018 29
Resolution Limit of Bragg Cells
Angular scan over the full bandwidth B
∆θ =λfuva− λfl
va=
λB
vaIlluminate the full aperture X uniformly. Due to diffraction the angular beam spread
I(θ) = sinc2X
λθ
1st zero θ0 =λX =⇒ δθ4dB = λ
X
Number of resolvable angles
N =∆θ
δθ=
λB
v1
X
λ= TaB
sinc LKz
sinc XKx
X
L
δθ
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 30
Optical wave in Crystal - optical ~k space
z-axis
~koθ
~ke
se
so
~k × (~k × ~E) + ω2µǫ ~E = 0
k2
n2− k20
k2x + k2y
n2e
+k2zn2o
− k20
= 0 |~k| = 2πn
λ
det[kikj − δijk2 + ωµǫij] = 0
1
n2e(θ)
=cos2 θ
n2o
+sin2 θ
n2e
Monochromatic planewave
Uniaxial case
Positive uniaxial crystalNegative uniaxial crystal
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 31
Optical Crystal Properties
Uniaxial Biaxial Optically Active
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 32
Anisotropic Bragg Cell
Anisotropic Bragg Cell
A
L
Acousto-optic Bandshape
DE
f
δε(r)E(r)
aK
ki
kd
FT δε(r)E(r)
Optical Momentum Surfaces
Anisotropic Bragg Cell
aK
ki
kd
FT δε(r)E(r)
n ω/co
n ω/ceOptical
MomentumSurface
0f
uf
lf
f2pm
1pmf
Acousto-optic Bandshapewith 3dB midband rippleDE
f0f1pmf
uflf
f2pm
B
A
L
δε(r)E(r)
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 33
Tangentially phase-matched Bragg cell
aK
ki
kd
FT δε(r)E(r)
Optical Momentum Surfaces
Acousto-optic Bandshape
DE
f
A
L
δε(r)E(r)
• Tangentially phase-matched diffraction gives Wide bandwidth and High efficiency.
• Quadratic enhancement in efficiency for narrower fractional BW.
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 34
Tangentially phase-matched Bragg cell
aK
ki
kd
FT δε(r)E(r)
Optical Momentum Surfaces
Acousto-optic Bandshape
DE
f
A
L
δε(r)E(r)
• Tangentially phase-matched diffraction gives Wide bandwidth and High efficiency.
• Quadratic enhancement in efficiency for narrower fractional BW.
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 35
Tangentially phase-matched Bragg cell
aK
ki
kd
FT δε(r)E(r)
Optical Momentum Surfaces
Acousto-optic Bandshape
DE
f
A
L
δε(r)E(r)
• Tangentially phase-matched diffraction gives Wide bandwidth and High efficiency.
• Quadratic enhancement in efficiency for narrower fractional BW.
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 36
Tangentially phase-matched Bragg cell
aK
ki
kd
FT δε(r)E(r)
Optical Momentum Surfaces
Acousto-optic Bandshape
DE
f
A
L
δε(r)E(r)
• Tangentially phase-matched diffraction gives Wide bandwidth and High efficiency.
• Quadratic enhancement in efficiency for narrower fractional BW.
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 37
Tangentially phase-matched Bragg cell
aK
ki
kd
FT δε(r)E(r)
Optical Momentum Surfaces
Acousto-optic Bandshape
DE
f
A
L
δε(r)E(r)
• Tangentially phase-matched diffraction gives Wide bandwidth and High efficiency.
• Quadratic enhancement in efficiency for narrower fractional BW.
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 38
Wider bandwidth through 3dB ripple design
aK
ki
kd
FT δε(r)E(r)
n ω/co
n ω/ceOptical
MomentumSurface
0f
uf
lf
f2pm
1pmf
Acousto-optic Bandshapewith 3dB midband rippleDE
f0f1pmf
uflf
f2pm
B
A
L
δε(r)E(r)
• Angularly detuned diffraction gives wider 3dB rippled bandwidth .
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 39
Wider bandwidth through 3dB ripple design
aK
ki
kd
FT δε(r)E(r)
n ω/co
n ω/ceOptical
MomentumSurface
0f
uf
lf
f2pm
1pmf
Acousto-optic Bandshapewith 3dB midband rippleDE
f0f1pmf
uflf
f2pm
B
A
L
δε(r)E(r)
• Angularly detuned diffraction gives wider 3dB rippled bandwidth .
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 40
Wider bandwidth through 3dB ripple design
aK
ki
kd
FT δε(r)E(r)
n ω/co
n ω/ceOptical
MomentumSurface
0f
uf
lf
f2pm
1pmf
Acousto-optic Bandshapewith 3dB midband rippleDE
f0f1pmf
uflf
f2pm
B
A
L
δε(r)E(r)
• Angularly detuned diffraction gives wider 3dB rippled bandwidth .
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 41
Wider bandwidth through 3dB ripple design
aK
ki
kd
FT δε(r)E(r)
n ω/co
n ω/ceOptical
MomentumSurface
0f
uf
lf
f2pm
1pmf
Acousto-optic Bandshapewith 3dB midband rippleDE
f0f1pmf
uflf
f2pm
B
A
L
δε(r)E(r)
• Angularly detuned diffraction gives wider 3dB rippled bandwidth .
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 42
Wider bandwidth through 3dB ripple design
aK
ki
kd
FT δε(r)E(r)
n ω/co
n ω/ceOptical
MomentumSurface
0f
uf
lf
f2pm
1pmf
Acousto-optic Bandshapewith 3dB midband rippleDE
f0f1pmf
uflf
f2pm
B
A
L
δε(r)E(r)
• Angularly detuned diffraction gives wider 3dB rippled bandwidth .
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 43
Wider bandwidth through 3dB ripple design
aK
ki
kd
FT δε(r)E(r)
n ω/co
n ω/ceOptical
MomentumSurface
0f
uf
lf
f2pm
1pmf
Acousto-optic Bandshapewith 3dB midband rippleDE
f0f1pmf
uflf
f2pm
B
A
L
δε(r)E(r)
• Angularly detuned diffraction gives wider 3dB rippled bandwidth .
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 44
Wider bandwidth through 3dB ripple design
aK
ki
kd
FT δε(r)E(r)
n ω/co
n ω/ceOptical
MomentumSurface
0f
uf
lf
f2pm
1pmf
Acousto-optic Bandshapewith 3dB midband rippleDE
f0f1pmf
uflf
f2pm
B
A
L
δε(r)E(r)
• Angularly detuned diffraction gives wider 3dB rippled bandwidth .
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 45
Phased Array Bragg Cell
+ − −+ +
K a+K a−
fbs
3fbs
fbs
3fbs
K a+K a−
+ − −+ +
Real SpaceTransducer Angular Radiation Pattern Momentum Space
Kbs
K ( )θa
K ( )θa
θ
θ
d
Kbs
Kbs
L
2 /Lπ
π2 /d
bsK = /dπ
Phased-Array Bragg Cell
aK
ki
kd
bsK
Acousto-optic Bandshapewith 3dB midband ripple
DE
f
Floating ground planePhased-array
transducer
A
L
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 46
Acoustic time-delay staircase phased arrayproduces only 1 beamsteered order (+3dB)
– just like a blazed grating
+−
+−
+−
+−
+−
+−
+−
+−
+−
K2
K3
K1ink
optical index surface
kout
beamsteering locus
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 47
Dielectric Perturbation incuced by a stair-casetime-delay phased array
Time-Delay Phased-Array
+−
+−
+−
Ein
h : step height
d : step width
s : step transducer width
A : optical aperture width
L : total interaction length
Graphical Representation
s
∆εe x
h
d
L
+−
+−
zA
Π ( )xA
Analytical Representation
∆ǫΩe (z, x) =
∫eikzzeikx(kz,Ω)xdkz
[∆ǫΩA
∫ ∏(zs
)e−ikzzdz
]
∆ǫΩr (z, x) = ∆ǫΩe (z, x) ∗
∏
(z
L) ·∑
q
(−1)qδ(z − qd)δ(x− qh)
·∏
(x
A)
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 48
Momentum Distribution
k x
θ
zk
acousticeigensurface@ Ω
zk
k x
2
01
zk
samplinggrids
πh
q =−1
( )2zk ssinc
2−D sincfunction
( )2zsinc k L
k A( )2
sinc x
elementfunction
K2
K3
K1ink
optical index surface
kout
beamsteering locus
Fxz∆ǫr ≈ ∆ǫΩ sinc
(kz −
√(Ω
Va(θ)
)2− k2x
)s
2
·∑
q
δ (hkx − dkz − (2q − 1)π)
∗[sinc
(kzL
2
)· sinc
(kxA
2
)]
Dielectric momentum distribution is the Fourier transform of dielectric perturbation.Acoustic diffraction is represented in the spectral domain
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 49
Time Delay Staircase Blazed Phased ArrayBragg Cell
kz
kx
K2
K3
1
2
3
sampling grid
zk
acousticeigensurfaces
2 πs
K1
kin
kout
KA
kin
kout
KA
kin
kout
KA
+−
+−
+−
+−
+−
+−
+−
+−
+−
Time-delay phased-array beamsteering tilts locus of momentum vectors to be tangentialto optical momentum surface thereby achieving wide bandwidth.
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 50
Acousto-optic modulator
Transit time of acoustic wave across focused beam profile of width d0 limits rise timeto τ = d0/Va. Need to minimize d0 or use large acoustic velocity crystal.
Incident
Diffracted∆θ=Λ/LIncident Diffracted
UndeflectedL
AOM
∆φ∆φ
d0
iθ
θd
AOM
s(t) DCblock
Real Space Fourier Space
GratingPeriod=Λ=v/f
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 51
Optimal alignment of Acousto-optic modulator
What is the optimum value of a = ∆φ∆θ ?
1 2 3 4
0.5
1.0
1.5
2.0
ηmax
τ /τr
η α L
τ increases with
beam widthr
Rise Time and DE vs a
a≪ 1, τr ≈ 0.65d0/V and ηmax≪ 1 fast but inefficient
a = 1.5, τr ≈ 0.85d0/V and ηmax ≈ 0.5 good balance between DE and rise time
a≫ 1, τr increases and ηmax→ 1 slow risetime but efficient
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 52
Alignment of beam divergence illuminatingAOM
H
MatchingNetwork
GroundPlane
Piezoelectric Transducer Top
Electrode
Angle cutback facet
TransparentPhotoeleastic
Medium
L
Beam should only be this big
Tightly Focused Beam Illuminating AOM
AOM
• Begin by using a tigtly focusing beam illuminating the AOM
• Note the overlap between the DC beam cone and diffracted cone
• Diffracted cone is modulated by a Bragg selective sinc in x
• stop down beam so that only central lobe is in beam
• Now note that DC is separated from diffracted cone
• realign to produce this divergence condition with full laser power
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 53
Physical Acoustics
Displacement vector field ~u(~r, t)Transverse Wave
Longitudinal Wave
tan S-1xytan 2S-1
xy
Simple Shear Pure Shear
tan S-1xy
Plane wave~u(~r, t) = Wu cos(Ωt− ~K · ~r) = u cos
[Ω
(t−
~M · ~rva( ~M)
)]
Displacement Gradient Matrix
Qij(~r, t) =
[∂ui(~r, t)
∂rj
]=
∂ux/∂x ∂ux/∂y ∂ux/∂z∂uy/∂x ∂uy/∂y ∂uy/∂z∂uz/∂x ∂uz/∂y ∂uz/∂z
Can be broken into symmetric and anti-symmetric parts
Sij =1
2
[∂ui∂rj
+∂uj∂ri
]Strain give restoring forces
ωij =1
2
[∂ui∂rj− ∂uj
∂ri
]Local rotations, no restoring forces
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 54
Displacement Gradient Matrix and Strain
1-D strainIn 1-D strain is ratio of increase in length to original length
P ′Q′ − PQ
PQ=
∆u
∆xe = lim
∆x→0
∆u
∆x=
du
dx
x
x+u
∆x
∆x+∆uO
P Q
O P’ Q’
2-D strainConsider the stretching of a piece of graph paperDeformation moves P = (x1, x2) to P ′= (x1+u1, x2+u2)and nearby Q=(x1+∆x1, x2+∆x2) toQ′=(x1+∆x1+u1+∆u1, x2+∆x2+u2+∆u2)
∆ui =∂ui∂xj
∆xj = eij∆xj ⇒ eij is a tensor
P Q1
Q2P’
Q’1Q’2
φ
Displacement in a rigid body rotation
P’
2-D displacements
P Q1
Q2
∆x1
12e 21e
∆u12∆u
Q’1Q’2
PQ1 ⇒ ∆x2 = 0 ⇒ ∆u1 = e11∆x1 ∆u2 = e21∆x1e11 measures the extension per unit length along 1, just like 1-De21 measures anticlockwise rotation of PQ1 tan θ = ∆u2
∆x1+∆u1≈ ∆u2
∆x1Does displacement gradient matrix measure strain?
Under a rigid body rotation, strain should be zero but eij =[0 −φφ 0
]
So decompose into symmetrical Strain and antisymmetrical local rotation parts
eij = Sij + ωij Sij =12(eij + eji) ωij =
12(eij − eji)
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 55
Pesky factors of 2
Convention is to include 2 in reduced subscript Strain but not Stress
Sij =12(eij + eji)
S11 S12 S13
S21 S22 S23
S31 S32 S33
=
e1112(e12 + e21)
12(e13 + e31)
12(e12 + e21) e22
12(e23 + e32)
12(e13 + e32)
12(e23 + e31) e33
=
S1
12S6
12S5
12S6 S2
12S4
12S5
12S4 S3
⇒
S1
S2
S3
S4
S5
S6
T1 T6 T5
T6 T2 T4
T5 T4 T3
⇒
T1
T2
T3
T4
T5
T6
Hooke’s law for crystals
Sij = sijklTkl ⇒ SI = sIJTJ Tij = cijklSkl ⇒ TI = cIJSJ [s] = [c]−1
Compliancessijkl = sMN when M and N are 1,2,32sijkl = sMN when either M or N are 4,5,64sijkl = sMN when both M or N are 4,5,6
Stiffnessescijkl = cIJ with no factors of 2
Piezoelectricdijk = 2diJ when J is 4,5,6
eg Txx = cxxyySyy, T1 = c12S2, T1 = Txx, S2 = Syy ⇒ c12 = cxxyy
Txy = cxyxySxy + cxyyxSyx = 2cxyxySxy, T6 = 2cxyxy
(S62
)= cxyxyS6 ⇒ c66 = cxyxy
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 56
Force per unit area acting on dV: Stress
Body forces are long range forces acting directly on all theparticles in a volume. eg Gravity ~FdV = ρ~gdVTraction forces instead are transmitted between neighbor-ing particles. Cartesian Force components (per unit area)on each face of a differential volume element
~Ti = xTix + yTiy + zTiz
Traction force on arbitrary face given by Tn=T · n=Tijnj
Stress components Tij are functions of spatial position, socontributions from each surface must be summed to giveforce acting on dV which for differential cube volume is
Unit volume element withedges parallel to principal stress directions
x
y
z
Tz
Ty
Tx
T33
T13
T23
T11
T12
T13
T32T12
T22
r
(r+dr,t)
∫
S
T·ndS=(T+x−T−x )x+(T+
y−T−y )y+(T+z−T−z )z =
∫
V
~FdV
where ~F is any force such as ma = m~u. In the limit ofsmall dV
~F = ∇ ·T = limdV→0
∫S T · ndS
dV
linear nonlinearplastic
deformation
Fra
ctur
e po
int
elastic deformation Stress
Str
ain Linear up to
about 1% S
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 57
Acoustic Wave Propagation
Generalize Hooke’s Lawrelates stress Tij (symmetric in non-feric) TIto strain Skl, reduced subscript SJ
Tij = cijklSkl TI = cIJSJ
cijkl compliance, 4th rank
symmetric in ij and in kl so can use reduced subscripts
I = 1, 2, 3, 4, 5, 6 for ij = 11, 22, 33, 23, 13, 12[
11 12 ← 13ց ↑
22 23ց ↑
33
]→
S1S2S3S4S5S6
energy arguments show cIJ = cJI
Acoustic Wave propagation couples S and T
Restoring force on displaced particles
~F = ∇ · T = ρm∂2~u
∂t2
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 58
Acoustic Wave Propagation 2
Fi =∂
∂xjTij = cijkl
∂
∂xjSkl = cijkl
∂2uk∂xj∂xl
= ρm∂2ui∂t2
Force = mass density times acceleration
For plane wave propagating along ~K at frequency Ω
cijklKjKluk = ρmΩ2ui
Christoffel characteristic equation for each unit direction M =~K
|~K|[cijklMjMl/v
2a(M)− ρmδik
]uk = 0
Solve by setting det[ ] = 0
3 solutions for each direction M
slowness1
va(M)=|~K|Ω
eigenpolarizations u(M)
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 59
Acoustic wave in Crystal - Acoustic ~K space
z
x
y
propagation direction
Shear acoustic wave
z
x
y
propagation direction
Shear acoustic wave
z
x
y
Uniform crystal volume element
propagation direction
z
x
y
Longitudinal acoustic wave
[cijklMjMl/V2a (
~M)− ρmδik]uk = 0
Christoffel characteristic equation
M ⇒
3 solutions for each propagation
1
Va(M)=| ~K|Ω
Slowness:
GaPDisplays the symmetry of the crystal43m
direction
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 60
Acoustic Velocity and Slowness surfaces forGaP
Self Collimated
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 61
Acoustic Velocity and Slowness surfaces forLiNbO3
Slowness surface crossections in 3-D to illustrate polarizations
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 62
TeO2 slowness surface
TeO2 slowness and velocity surfaces in and perpendicular to AO planeSlowness Surfaces
Velocity Surfaces
Ka
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 63
TeO2 eigenpolarizations in the presence ofOptical Activity
LiNbO3 Conoscopic Pattern
TeO2 Conoscopic Pattern[110]
Optically Rotated
Optically Unrotated
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 64
Diffraction Efficiencies for off-axis TeO2:
the photo-elastic tensor p and effective peff
peff = po ǫpS ǫ pi p = pIJ =
p11 p12 p13 0 0 0p12 p11 p13 0 0 0p31 p31 p33 0 0 00 0 0 p44 0 00 0 0 0 p44 00 0 0 0 0 p66
p11 = .633
p12 = .0074
p13 = .34
p31 = .0905
p33 = .24
p44 = −.017p66 = −.0463
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 65
Crystal Orientation-TeO2 momentum surface
TeO2 Momentum surface with Slowness surface
• TeO2 is positive uniaxial
• TeO2 is optically active
• Tangential birefringent diffraction• Polarization switching geometry
• θa Acoustic rotation changes fo
• θo Optical rotation changes fo
• High acoustic walk-off angle with θa
•Midband degeneracy when θa=0
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 66
Optical and Acoustic rotation
KA(Ω2)
Acoustic Walk-off
SA
ko(ω1)
ke(ω1)
Acoustic rotation
Elliminate midband degeneracy
Increase center frequency
Acoustic rotation in optically rotated plane
Allows center frequency to be tuned
Elliminate degeneracy throughout BW
Optical rotation
Midband degeneracy
k-space slice
KA(Ω2)
ke(ω1)
θ Optically Rotated Sliceo
ko(ω1)
KA(Ω1)
ke(ω1)
[001]
Unrotated Slice
ko(ω1)
KA(Ω1)
ke(ω1)
ko(ω1) [001]
SA
Acoustic Walk-off
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 67
Schaeffer-Bergman Experiments
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 68
Schaeffer-Bergman k-space intersectiongeometry for α-BBO
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 69
Schaeffer-Bergman results and fit for α-BBO
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 70
Optical Rotation in a Uniaxial CrystalWith No Optical Activety
• Tangentially degenerate frequency goes to 0 along z in absence of optical activity
•Maximum frequncy f = Vf
√n2h − n2
l
– V depends on acoustic wave direction in x, y plane
• Eigenmodes of ordinary input and extraordinary diffraction not perpendicular!
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 71
Inband degeneracy - Acoustic rotation
aK
ki
kd
FT δε(r)E(r)
n ω/co
n ω/ceOptical
MomentumSurface
0f
f2pm
1pmf
2 degree Acoustic rotation
3 degree Acoustic rotation
• Degenerate diffraction reduces the usable bandwidth of AOD.– 2 rotation still has degeneracy in band.– 3 rotation sufficient to elliminate degenerate diffraction.
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 72
Transducer Length, Height, and Shape
Patterned Electrode
PiezoelectricTransducer
Incident Beam size A
Acoustic Wave
Diffracted Beam
[001]
K space
Acoustic
Vector Ka
2πL
2πA
Uncertainty Box
Crystal OrientationCrystal SizeElectrode ShapeElectrode Size
Patterned Electrode
L
H
Piezoelectric Transducer
Plane of incidence
Crystal
• Transducer can be Longitudinal particle motion or Transverse.
– Shear transducer must be oriented to excite desired shear wave.
– Misorientation can excite other shear. Extraneous diffraction unless peff = 0
• Transducer Length determined by bandwidth requirement through Bragg matching
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 73
Transducer shape
H
L
• Diamond shape is preferred for its acoustic field uniformity distance.
• Elliminates vertical diffraction sidelobes in scanner Fourier plane (26dB vs 13dB)
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 74
Comparison of momentum-space andexperiment for Diamond Transducer
L = 2.9 mm
2.2425•1072.2430•1072.2435•107
KZ [m-1]
0
2•105
4•105
6•105
8•105
K11
0 [m
-1]
L = 2.3 mm
2.2425•1072.2430•1072.2435•107
KZ [m-1]
0
2•105
4•105
6•105
8•105
K11
0 [m
-1]
L = 1.9 mm
2.2425•1072.2430•1072.2435•107
KZ [m-1]
0
2•105
4•105
6•105
8•105
K11
0 [m
-1]
L = 2.9 mm
50 60 70 80 90 100F [Mhz]
0.0
0.2
0.4
0.6
0.8
1.0
η
ExperimentTheory
L = 2.3 mm
50 60 70 80 90 100F [Mhz]
0.0
0.2
0.4
0.6
0.8
1.0
η
ExperimentTheory
L = 1.9 mm
50 60 70 80 90 100F [Mhz]
0.0
0.2
0.4
0.6
0.8
1.0
η
ExperimentTheory
Calculation and measurement of diffraction efficiency of an acoustically apodized TeO2
Bragg cell for three effective rectangular transducer lengths. The k-space diagramsat band-center (f=72 Mhz) are shown on the top, band-shapes vs frequency on thebottom.
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 75
Elastic Energy and WalkoffIntrinsic symmetries cijkl = cjikl = cijlk = cklij
yx yx yx
Elastic Energy Density (like 12~E · ~D = 1
2EiǫijEj in E&M)
UA = 12TklSkl =
12cijklSijSkl =
12cijkl
∂ui∂xj
∂ul∂xk
Remember defn StrainSij =
1
2
(∂ui∂xj
+∂uj∂xi
)
Tij = cijklSkl =12cijkl
∂uk∂xl
+ 12cijkl
∂ul∂xk
= cijkl∂ul∂xk
yx
Wave Eqn
ρ∂2ui∂t2
=∂Tij
∂xj= cijkl
∂
∂xj
∂ul∂xk
Plane wave with unit displacement polarization p and unit direction n with velocity v
ui = ApiF
(t− n · ~x
v
)= ApiF
(t− njxj
v
)
ui = ApiF′
∂ul∂xk
= −nk
vAplF
′
ui = ApiF′′ ∂2ul
∂xj∂xk=
njnk
v2AplF
′′
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 76
Acoustic Walkoff
Plug in wave eqnρApi
ZZZF ′′ = cijklnjnkplA
ZZZF ′′/v2
ρv2︸︷︷︸ pi = cijklnjnk︸ ︷︷ ︸ pleigenvalue Γil
ρv2
|Γil −︷︸︸︷λ δil| = 0
dot with Api· to get Energy, and remember |p|2 = p2i = 1
A2ρv2p2i = cijklpinjnkplA2
UA = 12cijkl
∂ui∂xj
∂ul∂xk
= 12cijkl(−ApinjF
′/v)(−AplnkF′/v)
=A2
2cijklpinjplnk
F ′2
v2=
A2
2ρp2iF
′2 =A2
2ρF ′2
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 77
Acoustic Poynting Vector
Harmonic Plane Wave
ui=ApiF (t−njxj/v)=A2pie
jΩ(t−nmxm/v)+cc= A2pie
jΩ(t−smxm)+cc= A2pie
j(Ωt−Kmxm)+cc
Poynting vector Pi = −T ∗ij∂uj∂t
= −cijkl∂ul∂xk
∂uj∂t
∂uj∂t
= ApjF′ =
A
2pj(jΩ)e
j(Ωt−Kmxm) + cc
∂ul∂xk
= −nk
vAplF
′ =A
2pl(−jKk)e
j(Ωt−Kmxm) + cc
Pi = cijklA2pj
nk
vplF
′2 = cijklA2
2pj Kk︸︷︷︸ plj
2Ω
Ωsk = Ωnk/vEnergy Velocity
V ei =
Pi
U=
A2cijklpjnkvplF
′2
A2ρp2iF′2 =
cijklpjnkplΩ2
ρvΩ2= vgi
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 78
Acoustic Group Velocity
Equivalent eqns using unit normal nj, slowness sj=njv [ sm], or wavevector Kj=Knj=
Ωsj=Ωnjv [ 1m]
cijklnjnluk = ρv2ui ⇒ vα, uαi
cijklsjsluk = ρui ⇒ 1
vα, uαi
cijklKjKluk = ρΩ2ui ⇒ Kα, uαidot with unit particle displacement eigenpolarization pi = uαi . note p
2i =
∑i p
2i = 1
pi · [cijklKjKlpk = ρΩ2pi] cijklpiKjpkKl = ρΩ2p2i
Ω2 =cijklpiKjpkKl
ρ
Now find ~vg = ∇ ~KΩ = ∂Ω∂Kl
l
∂Ω2
∂Kl=
dΩ2
dΩ
∂Ω
∂Kl= 2Ω
∂Ω
∂Kl⇒ ~vg =
∂Ω
∂Kll =
1
2Ω
∂Ω2
∂Kll
vgl =1
2Ω
∂
∂Kl
[cijklpiKjpkKl
ρ
]=
1
2Ω2cijklpiKjpk
ρ=
cijklpinjpk
ρΩK
=cijklpinjpk
ρv=
cijklpisjpkρ
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 79
Acoustic Walkoff in Acoustically Rotated TeO2Top view: acoustic face
θWO=27o
θAR=2.6o
k
s[1
10
]
23.64mm RectangularPiezoelectric Transducer
Length = 45 mm
Width =42 mm
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 80
AOTF FundamentalsT e r m i n a t i o n
R F
D i f f r a c t e d b e a m
C o l l i n e a r C a M o O 4 A O T F
P B S
z
x k i y k d
x k a
K - s p a c e o f c o l l i n e a r A O T F
y
λ
DE f=f1
λ
DE f=f2
λ
DE f=f3
λ 1
λ 2
λ 3
~k2 = ~k1 + ~Ka =⇒ n2ω + Ω
c= n1
ω
c+
Ω
v=⇒ Ω
ω≈ |n2 − n1|
v
c
• RF signal applied to piezo-electric transducer launches an acoustic shear wave
photoelastic coupling produces off-axis pertubation to dielectric tensor
• Each RF frequency component couples a specific wavelength from o to e
Polarizer selects diffracted component
• Octave RF and optical bandwidth with sub nm resolution
• Optical frequency Doppler shifted by RF frequency hνd = h(νo + νA)
• RF spectrum produces modulated optical spectrumS. E. Harris et al, “Electronically Tunable Acousto-Optic Filter , Vol. 15, No. 10, p325-326, Appl. Phys. Lett, (1969)
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 81
Non-Colinear Acousto-OpticTunable Filters
Apodized Noncolinear Acousto-optic Tuneable Filter (AOTF)
RFinput
Piezoelectric
Transducer
o-polarized
Input
Beam
e-polarized
diffracted
beam
OA
AcousticWave
Polarizer
Momentum Space Representation
K (Ω ) 1A
ParallelTangents
AcousticWalkoff
SA
ek (ω ) 1k (ω ) o 1
ek (ω ) 22k (ω ) o
2K (Ω ) A
• k-space for anisotropic optics and acoustics
– Anisotropic acoustic walkoff
– Colinear optical mode power flow
• Parallel tangents geometry
– Gives wide angular aperture
– Equal curvature at 55
• Tradeoff of DE vs Resolution
– Complicated by Optical Activity in TeO2
• Each RF frequency matched to ω∆n(ω)c
– Couples one wavelength from o to e
• Polarizer selects diffracted componentI. C. Chang, Acoustooptic Tunable Filters, Optical Engineering vol. 20, p. 824-829, 1981.
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 82
TeO2 Noncolinear AOTFsparallel tangents at various angles
tradeoff resolution with DE (Opt Act ×10)
R F
N o n c o l l i n e a r T e O 2 A O T F
l 1
P o l a r i z e r
K - s p a c e o f n o n c o l l i n e a r A O T F
z
x k i
e
k d e
k a
l
l
ka
k ikd
Parallel tangents condition
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 84
Parallel Tangents Condition:Use for Wide Angular Aperture AO
DiffractionI.C. Change, Acousto-Optic Tunable Filters, Opt.Eng., vol. 20(6), p. 824, 1981
Parallel Tangents Condition⇒ ~Se ‖ ~So
For uniaxials θo = θe + δ φo = φe
tan θo =n2o
n2e
tan θe
Derivation
Walkoff angle. Expand n2e(θe), write in terms of tangents
θe
θo
δθe
ke
ko
SeSo
ne(θ)ne no
z
x
θe
θa
KA
tan δ = n2e(θe)
sin 2θe2
[1
n2e
− 1
n2o
]=
sin θe cos θe[cos2 θen2o
+ sin2 θen2e
]n2o − n2
e
n2on
2e
=(n2
o − n2e) tan θe
n2e + n2
o tan2 θe
Take tangent of both sides of θo = θe + δ and use tangent sum formula
tan θo = tan(θe + δ) =tan θe + tan δ
1− tan θe tan δ=
tan θe +(n2o−n2e) tan θen2e+n2o tan
2 θe
1− tan θe(n2o−n2e) tan θen2e+n2o tan
2 θe
=tan θe(n2
e + n2o tan
2 θe) + (n2o −
n2e) tan θe
n2e +
XXXXXXXXXn2o tan
2 θe − (@@@n2o − n2
e) tan2 θe
= tan θen2o
n2e
(tan2 θe + 1)
(1 + tan2 θe)=
n2o
n2e
tan θe
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 85
Dispersion in TeO2 and nonlinearfrequency map in an AOTF
Dispersion of both no, ne and ρz, ρ⊥
• Strong dispersion of ∆n in visible
• NIR operation much more linear ν vs f
Allows Doppler shifting ν ∝ f
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 86
Quasi-Colinear AOTF (~ko ‖ ~Sa): colinear powerflow gives high resolution
61.6MHz @1064
84.1MHZ@800
137.58MHz@532
110
001
110
001
α=40.1
θa=
4.8
96
v=2100m/s
v=
61
5m
/s
18.57
φ=45
57.1
18.5745
slow shear transducerparticle motion perp to plane
110
45
110
001
Diffracted, pol switched spectrally filtered & pulse shaped output.Beam offset in position and angle
ko
ke
Ka1
Pa
Ka0
45 cut quasi-colinear AOTFo
111111
110AcousticSlowness
optical k-space
TeO2 AODCrystal Cut
• Acoustic reflection off boundary arranged to give desired mode with ~Sa ‖ nV. Voloshinov, Close to collinear acousto-optical interaction in paratellurite, Opt. Eng. v. 31(10), p 2089, 1992
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 87
Simple AO Device model
a(t) = cos 2πft =ei2πft + e−i2πft
2Signal on carrier and analytic representation
s(t) = a(t) cos 2πf0t = s(t) + s∗(t)
s(t) =e−i2π(f0+f)t + e−i2π(f0−f)t
4Signal diffracted by AOD
d(x, t) = w(x)s(t− x/v)e−i(ωt−~k·~r)
= w(x)e−i2π(f0+f)(t−x/v) + e−i2π(f0−f)(t−x/v)
4e−i(ωt−
~k·~r)
= w(x)e−i2πf0(t−x/v)
2
[e−i2πf(t−x/v) + ei2πf(t−x/v)
2
]e−i(ωt−
~k·~r)
= w(x)e−i2πf0(t−x/v)
2cos 2πf(t− x/v)e−i(ωt−
~k·~r)
Detected IntensityI(x, t) =
|w(x)|24
cos2 2πf(t− x/v)
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 91
Simple AO Device model 2
Wideband Input can be Fourier Decomposed
d(x, t) = Aiǫ
2Π
[x−X/2
X
]e−i2πνt
∫ 0
−∞S(f)ei2πf(t−x/v)df
= Aiǫ
2Π[x−X/2
X
] ∫S(f)ei2π[(f−ν)t−fx/v]df = A
iǫ
2Π[x−X/2
X
]e−i2πνts(t− x/v)
Simple model
d(x, t) = AΠ
[x−X/2
X
]s(t− x/v)
Window Function
w(x) = Π
[x−X/2
X
]e−(x−x0−X/2)2/σ2e−α(f)x/2a(x, y)
X = crystal width (5-50mm)
σ = Illuminating Gaussian Width
x0 shifted off center
α(f) = α0 + f 2α2 Quadratic Frequency dependent acoustic attenuation (in crystals )
a(x, y) Projected through z transducer diffraction pattern
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 92
Acousto-optic Spectrum Analyzer
AOD
s(t)
S(xv/Fλ)
s(t-x/v)
F
CCD
At the back focal plane the field is
S(x′, t) =
∫
A
s(t− x/v)e−iω(t−x/v)e−i2πxx′/λFdxe−i2πνt
Temporal Integration gives the power spectrum
I(x′) =
∫ T
0
|S(x′, t)|2dtNumber of resolvable frequency bins TB = 1000Distributed Local Oscilator (DLO) DLO = e−ivx
′t/λF
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 93
AO spectrum analyzer analysis
A(u, t) =
∫w(x)s(t− x
v)e−i2ıuxdxe−i2πνt
=
∫w(x)
∫H(f)S(f)ei2πf(t−x/v)dfe−i2ıuxdxe−i2πνt
=
∫H(f)S(f)ei2πftW
(u− f
v
)dfe−i2πνt
=[vH(uv)S(uv)ei2πvtue−i2πνt
]∗W (uv)
H(f) AOD frequency response
w(x) AOD Window
W (u) =
∫w(x)e−i2ıuxdx impulse response in the Fourier plane
ei2πvtu Distributed Local Oscillator (DLO). Rocking plane wave pivoted at DC
I(u, t) = |A(u, t)|2 = |vH(uv)S(uv) ∗W (uv)|2
I(u)
∫ T
0
|vH(uv)S(uv) ∗W (uv)|2dt
where u = x′/λF . spectral resolution = width W (uv)
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 94
Hybrid Window Function
w(x) = Π( x
X
)e−(x−x0)
2/σ2e−α(f)(x+X/2)eiφ(x)
X = AOD Widthσ = 1/e width of Gaussian illumination
x0 = offset of illumination from center
α(f) = α0f2 frequency dependent attenuation
φ(x) phase response:acoustic diffraction, optical imperfections
Width of convolution ≈ 1X + 1
πσ +α2π
e−x2/σ2e−αx = e−(x+
12ασ
2)2/σ2e14α
2σ2
completing the square shows that fixed attenua-tion is corrected by shifting the Gaussian towardsthe weaker side
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 95
Truncated Gaussian Impulse Response
w(x) = Π
(x
D− 1
2
)e−4T
2(x/D−.5)2
σ = 12T = ω0
D , truncation ratio T = D2ω0
, 2ω0 = 1/e2 intensity width.
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 96
Variation of peak heigth, width, and sidelobewith truncation ratio
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 97
Interferometric Detection
s(t)=cos(2πft)
SignalAOD
|S+r |20Reference
Beam
cos(2πft)
I(u, t) =∣∣A(u, t) + r0e
i2πνt∣∣2
=∣∣S(uv) ∗W (uv)
∣∣2 + |r0|2 + 2r0|S(uv)| cos [2πvtu + ∠S(uv)] ∗W (uv)
for a single tones(t) = |a| cos(2πf ′t + φ)
I ′f(u, t) =∣∣∣ax|W (u− f ′/v)ei(2πf
′t+φ)ei2πνt + r0ei2πνt
∣∣∣2
= |a|2W 2(u− f ′/v) + |r0|2 + 2|a|r0W (u− f ′/v) cos(2πf ′t + φ)
Temporal modulation on the last term reproduces the input sinusoid in frequency,amplitude, and relative phase.
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 98
2-Tone Intermods and Thermal noise floor
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 99
Intermodulation Products
f f
2f 2f
2f -ff -f
1 2
1 2
1 2 2 11 22 1f -f 2f -f
f +f 1 2
0
Optical Multiple Difftraction Orders
Acoustic Nonlinearities
Amplifier Nonlinearities
Coupled Mode theory indicates that all possible multiple sum and difference frequenciescan be generated.
In order to elliminate strong second order terms we must be limited to less than anoctave bandwidth.
Third and higher order intermodulations fall within band and limit DR
Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 100