Acousto-optic modulators and de ectors Historical overview...

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Acousto-optic modulators and deflectors An RF signal is applied to an impedance matching network that is connected across the electrodes of a thin piezoelectric transducer that is bonded to a photoelastic material (oriented, cut, polished crystal or isotropic glass) which launches an acoustic wave into the interaction medium. The acoustic wave induces a traveling-wave volume dielectric grating pertubation through the photoelastic effect which can efficiently diffract an appropriately 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 piezoelectric transducers, 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 Δǫ = ε 0 W 2 ǫpSǫe it k n ·x) + cc E m (x,t)= ˆ e m A m (z)e i(ω m t k m ·x) + cc Conservation of Energy photon phonon E = ω m E = Ω ω 0 ± Ω= ω 1 ω 0 1 2π.5 × 10 15 rad/sec Ω 2π10 9 rad/sec = λ 0 λ 1 Conservation of Momentum photon phonon p m = h k m p A = h K A k 1 = k 0 ± K A | k m | = 2π λ m /n = ⇒| k 0 |≈| k 1 | k 1 k 0 K A Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 3 Bragg condition sin θ B = λ/2 Λ Bragg Angle θ B θ B K G k d k i 2πn λ Λ θ B λ/2 λ/2 Λ θ B Λ= v a f Grating spacing = velocity frequency θ B = sin 1 | K| 2| k| = sin 1 λ 2nΛ = sin 1 λf 2v a n λf 2v a n Angle frequency Kelvin Wagner, University of Colorado Advanced Optics Lab 2018 4

Transcript of Acousto-optic modulators and de ectors Historical overview...

Page 1: Acousto-optic modulators and de ectors Historical overview ...ecee.colorado.edu/~ecen5606/VUGRAPHS/aolab18-nup.pdf · Acousto-optic modulators and de ectors An RF signal is applied

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

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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

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

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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

)

∆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

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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

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

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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

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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

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

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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

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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

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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

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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

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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

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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

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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

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

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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

(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

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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

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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

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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

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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

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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

Page 20: Acousto-optic modulators and de ectors Historical overview ...ecee.colorado.edu/~ecen5606/VUGRAPHS/aolab18-nup.pdf · Acousto-optic modulators and de ectors An RF signal is applied

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

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

∂Ω

∂Kl= 2Ω

∂Ω

∂Kl⇒ ~vg =

∂Ω

∂Kll =

1

∂Ω2

∂Kll

vgl =1

∂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

Page 21: Acousto-optic modulators and de ectors Historical overview ...ecee.colorado.edu/~ecen5606/VUGRAPHS/aolab18-nup.pdf · Acousto-optic modulators and de ectors An RF signal is applied

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

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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

[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

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

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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

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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