Workshop: Opto-acoustics with COMSOL - CUDOScudos.org.au/nusod_tutorial.pdf · Workshop: Opto-acoustics with COMSOL Christian Wol 1;3 and Michael J. A. Smith 2 1: Centre for Ultrahigh

Workshop: Opto-acoustics with COMSOL

Christian Wolff1,3 and Michael J. A. Smith1,2

1. Centre for Ultrahigh bandwidth Devices for Optical Systems (CUDOS),2. Institute of Photonics and Optical Science (IPOS), School of Physics, University of

Sydney,3. School of Mathematical and Physical Sciences, University of Technology Sydney

July 15, 2016

C. Wolff & M. J. A. Smith Workshop: Opto-acoustic with COMSOL July 15, 2016 1 / 21

Outline

Introduction

Acoustic waveguide problem

Calculation of an electrostrictive SBS gain


Outline

Introduction




Outline

Introduction




Outline

Introduction




Aims

Crash course in continuum mechanics

How to formulate custom PDEs in FEM solvers

How to couple light and sound fields

We use Brillouin as an example for opto-mechanical coupling.

We restrict ourselves to electrostrictive coupling.

Non-Brillouin and non-electrostrictive problems (e. g. cavityopto-mechanics or MEMS) require fundamentally similarmethods.


Aims








Aims








Aims








Aims








Aims








Brief intro to stimulated Brillouin scattering

Resonant non-linear acousto-optical process:

Sound wave: very shallow traveling grating

Doppler effect: scattered light shifted in ω; different β

Missing energy and momentum → sound wave amplification

Typical frequency shift: 1–10GHz

Massive difference in optical and acoustic frequencies:sound quasi-static for light field









pumpwaveguide









pumpwaveguide









sound wave transmittance

pump









sound wave transmittance

pump









sound wave

pump wavelength

Stokes wavelength

transmittancereflected spectrum









sound wave

pump wavelength

Stokes wavelength










sound wave

pump wavelength

Stokes wavelength










sound wave

pump wavelength

Stokes wavelength



Typical description: coupled mode theory (1/2)

Light oscillates 105 times faster than sound

Clearly: only slowly varying light signals relevant

Introduce fields as modulated eigenmodes:

E(r, t) = a1(z , t)e1(x , y , t) + a2(z , t)e2(x , y , t) + c.c.

e1(x , y , t) & e2(x , y , t): optical waveguide modes

a1(z , t) & a2(z , t): slowly varying envelopes

Analogously introduce acoustic field

U(r, t) = b(z , t)u(x , y , t) + c.c.

u(x , y , t): mechanical displacement field (explained in part 2).

b(z , t): acoustic envelope.










U(r, t) = b(z , t)u(x , y , t) + c.c.












U(r, t) = b(z , t)u(x , y , t) + c.c.












U(r, t) = b(z , t)u(x , y , t) + c.c.












U(r, t) = b(z , t)u(x , y , t) + c.c.












U(r, t) = b(z , t)u(x , y , t) + c.c.





Equations of motion for envelopes (steady state):

∂za1 = −iωQP−11 a2b

∗

∂za2 = −iωQ∗P−12 a1b

∂tb + αb = −iΩQE−1b a∗1a2

Problem defined by:

modal powers P1, P2 and energy Ebacoustic loss parameter α = Ω/QF

opto-acoustic coupling coefficient Q

Experimentally most relevant:

SBS power gain Γ = 2ω|Q|2QF/(P1P2Eb)





∗



Problem defined by:

modal powers P1, P2 and energy Eb

acoustic loss parameter α = Ω/QF








∗



Problem defined by:









∗



Problem defined by:









∗



Problem defined by:






Task of this workshop

Compute SBS power gain

Γ = 2ω|Q|2QF/(P1P2Eb)

of simple rectangular waveguide from:

modal power P1 = P2 and modal energy Ebopto-acoustic coupling coefficient Q

mech. quality factor QF = 1000 (assumed)

Required steps:

Set up optical waveguide problem and find mode

Set up acoustic waveguide problem and find mode

Numerically compute three missing integrals

Your turn: optical waveguide problem






modal power P1 = P2 and modal energy Eb



Required steps:












Required steps:












Required steps:












Required steps:












Required steps:












Required steps:












Required steps:








Fundamentals of continuum mechanics:

Two main observables:

deformationcorresponds to position of point masses

momentum densitycorresponds to momentum of point masses

Goal of this introduction:

PDE for continuum-mechanical waveguide problem






























Deformation of solid bodies

Intuitive key quantity:mechanical displacementfield u(r)

Only relevant: relativedisplacements,i. e. deformation of unit cells

Irrelevant constants can beremoved by differentiation(e. g. forces vs. potentials)

Define strain field S(r) assymmetrized gradient of u(r):

Sij = 12

(∂uj∂xi

+ ∂ui∂xj

); S = grad su







Sij = 12

(∂uj∂xi

+ ∂ui∂xj

); S = grad su







Sij = 12

(∂uj∂xi

+ ∂ui∂xj

); S = grad su







Sij = 12

(∂uj∂xi

+ ∂ui∂xj

); S = grad su







Sij = 12

(∂uj∂xi

+ ∂ui∂xj

); S = grad su







Sij = 12

(∂uj∂xi

+ ∂ui∂xj

); S = grad su

normal strain

normal strain

shear strain


Momentum in solid bodies (1/2)Classical momentum of pointmass: P = m ∂<u>

∂t.

Conserved (extensive)quantities in continuousproblems appear as densities;e. g. mass density ρm(r) orcharge density ρc(r).

Classical momentum density

in solid: p(r) = ρm(r)∂u(r)

∂t.

All conserved densities fulfill acontinuity equation:∂tρ + div j = 0

point mass:



∂t.




∂t.


point mass:



∂t.




∂t.


point mass:

center of mass:

solid body:



∂t.




∂t.


point mass:

center of mass:

solid body:



∂t.




∂t.


point mass:

center of mass:

solid body:


Momentum in solid bodies (2/2)

Momentum is a conserved density⇒ has associated current density

Each component of p has a vectorial current⇒ “momentum current” is 2nd rank tensor.

The “momentum current” is called stress tensor T:∂tp− divT = 0 .

Constitutive relation (Hook’s law):

Stress is a response of solid to deformation: T = T(S).

Simplest case: linear response Tij =∑kl

cijklSkl ; T = c : S .

c is called stiffness tensor.






























































Voigt notation

Fourth-rank tensors (e. g. the stiffness) are awkward.

Engineers prefer a compressed notation (Voigt notation) thatalso reflects the symmetry of S and T:

S =

S1 S6 S5

S6 S2 S4

S5 S4 S3

, T =

T1 T6 T5

T6 T2 T4

T5 T4 T3

,

S and T can be organized as “vectors” with 6 entries

This allows to represent c as a 6× 6-matrix

For isotropic materials c has only two free parameters, e. g.Youngs modulus E andPoisson number νand c contains many zeros.


Voigt notation



S =

S1 S6 S5

S6 S2 S4

S5 S4 S3

, T =

T1 T6 T5

T6 T2 T4

T5 T4 T3

,





Voigt notation



S =

S1 S6 S5

S6 S2 S4

S5 S4 S3

, T =

T1 T6 T5

T6 T2 T4

T5 T4 T3

,





Voigt notation



S =

S1 S6 S5

S6 S2 S4

S5 S4 S3

, T =

T1 T6 T5

T6 T2 T4

T5 T4 T3

,





Voigt notation



S =

S1 S6 S5

S6 S2 S4

S5 S4 S3

, T =

T1 T6 T5

T6 T2 T4

T5 T4 T3

,





Mechanical equations of motionCombine these equations:

∂tp− divT = 0

p = ρ∂tu

T = c : grad su = c : gradu(second step because of symmetries in c)

to find 3d mechanical wave equation:

ρ∂2t u− div (c : grad su) = 0

Assume z-propagating wave u(x , y , z) = u(x , y)eiβz

to find wave guide problem:

ρ∂2t (ux , uy , uz)− (∂x , ∂y ) ·

(T1 T6 T5

T6 T2 T4

)− iβ(T5,T4,T3) = 0 .



∂tp− divT = 0

p = ρ∂tu






ρ∂2t (ux , uy , uz)− (∂x , ∂y ) ·

(T1 T6 T5

T6 T2 T4

)− iβ(T5,T4,T3) = 0 .



∂tp− divT = 0

p = ρ∂tu






ρ∂2t (ux , uy , uz)− (∂x , ∂y ) ·

(T1 T6 T5

T6 T2 T4

)− iβ(T5,T4,T3) = 0 .



∂tp− divT = 0

p = ρ∂tu






ρ∂2t (ux , uy , uz)− (∂x , ∂y ) ·

(T1 T6 T5

T6 T2 T4

)− iβ(T5,T4,T3) = 0 .


Continuum mechanics summaryAcoustic waveguide problem:

ρ∂2t (ux , uy , uz)− (∂x , ∂y ) ·

(T1 T6 T5

T6 T2 T4

)− iβ(T5,T4,T3) = 0.

Stress = “momentum current” ⇒ zero-flux = “no-force”

Strain: Stress in isotropic material:

Sij =1

2

(∂uj∂xi

+∂ui∂xj

)

S =

S1 S6 S5

S6 S2 S4

S5 S4 S3

T1T2T3T4T5T6

=

c11 c12 c12 0 0 0c12 c11 c12 0 0 0c12 c12 c11 0 0 00 0 0 c44 0 00 0 0 0 c44 00 0 0 0 0 c44

S1S2S3

2S42S52S6

c11 =(1−ν)E

(1+ν)(1−2ν), c12 = νE

(1+ν)(1−2ν), c44 = E

2(1+ν)

Your turn: Implement and solve



ρ∂2t (ux , uy , uz)− (∂x , ∂y ) ·

(T1 T6 T5

T6 T2 T4

)− iβ(T5,T4,T3) = 0.



Sij =1

2

(∂uj∂xi

+∂ui∂xj

)

S =

S1 S6 S5

S6 S2 S4

S5 S4 S3

T1T2T3T4T5T6

=

c11 c12 c12 0 0 0c12 c11 c12 0 0 0c12 c12 c11 0 0 00 0 0 c44 0 00 0 0 0 c44 00 0 0 0 0 c44

S1S2S3

2S42S52S6

c11 =(1−ν)E

(1+ν)(1−2ν), c12 = νE

(1+ν)(1−2ν), c44 = E

2(1+ν)




ρ∂2t (ux , uy , uz)− (∂x , ∂y ) ·

(T1 T6 T5

T6 T2 T4

)− iβ(T5,T4,T3) = 0.


Strain:

Stress in isotropic material:

Sij =1

2

(∂uj∂xi

+∂ui∂xj

)

S =

S1 S6 S5

S6 S2 S4

S5 S4 S3

T1T2T3T4T5T6

=

c11 c12 c12 0 0 0c12 c11 c12 0 0 0c12 c12 c11 0 0 00 0 0 c44 0 00 0 0 0 c44 00 0 0 0 0 c44

S1S2S3

2S42S52S6

c11 =(1−ν)E

(1+ν)(1−2ν), c12 = νE

(1+ν)(1−2ν), c44 = E

2(1+ν)




ρ∂2t (ux , uy , uz)− (∂x , ∂y ) ·

(T1 T6 T5

T6 T2 T4

)− iβ(T5,T4,T3) = 0.



Sij =1

2

(∂uj∂xi

+∂ui∂xj

)

S =

S1 S6 S5

S6 S2 S4

S5 S4 S3

T1T2T3T4T5T6

=

c11 c12 c12 0 0 0c12 c11 c12 0 0 0c12 c12 c11 0 0 00 0 0 c44 0 00 0 0 0 c44 00 0 0 0 0 c44

S1S2S3

2S42S52S6

c11 =(1−ν)E

(1+ν)(1−2ν), c12 = νE

(1+ν)(1−2ν), c44 = E

2(1+ν)




ρ∂2t (ux , uy , uz)− (∂x , ∂y ) ·

(T1 T6 T5

T6 T2 T4

)− iβ(T5,T4,T3) = 0.



Sij =1

2

(∂uj∂xi

+∂ui∂xj

)

S =

S1 S6 S5

S6 S2 S4

S5 S4 S3

T1T2T3T4T5T6

=

c11 c12 c12 0 0 0c12 c11 c12 0 0 0c12 c12 c11 0 0 00 0 0 c44 0 00 0 0 0 c44 00 0 0 0 0 c44

S1S2S3

2S42S52S6

c11 =(1−ν)E

(1+ν)(1−2ν), c12 = νE

(1+ν)(1−2ν), c44 = E

2(1+ν)





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Simple picture:

Matter: many polarizable particles

P-field: Sum of individualpolarizations

Compression: Density increases

Higher P-field

Higher εr (Clausius-Mosotti)

More general description: linear change of εr -tensor due to strain field

Phenomenological description: Pockels tensor pijkl

∆εij = −ε0ε2r pijklSkl


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Simple picture:




Higher P-field






Photoelastic effect

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Simple picture:




Higher P-field






Photoelastic effect

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Simple picture:




Higher P-field






Photoelastic effect

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Simple picture:




Higher P-field






Photoelastic effect

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Simple picture:




Higher P-field






Photoelastic effect

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Simple picture:




Higher P-field






Electrostriction

Simple picture:


Intensity gradient(e.g. waveguide mode)

Gradient force

Body is compressed

Energetically linked to photoelasticity


Tij = ε0ε2r pijklEkEl


Electrostriction

Simple picture:



Gradient force

Body is compressed





Electrostriction

Simple picture:



Gradient force

Body is compressed





Electrostriction

Simple picture:



Gradient force

Body is compressed





Electrostriction

Simple picture:



Gradient force

Body is compressed





Electrostriction

Simple picture:



Gradient force

Body is compressed





Opto-acoustic coupling between modes

Coupled mode equations:∂za1 = −iωQP−1

1 a2b∗



Coefficient for excitationof acoustic mode: Qb = 〈s|T(ES)〉 =

∫Ω

d2r(s∗ : T(ES)

)Coefficient for scatteringbetween optical modes: Q1 = 〈e1|∆εr |e2〉 =

∫Ω

d2r e∗1 · [∆εr (S)e2]

In fact, bothare the same: Q = Q1 = Q∗

b =∑ijkl

∫Ω

d2r [e2]∗i [e1]jpijklskl




1 a2b∗




∫Ω

d2r(s∗ : T(ES)

)

Coefficient for scatteringbetween optical modes: Q1 = 〈e1|∆εr |e2〉 =

∫Ω

d2r e∗1 · [∆εr (S)e2]


b =∑ijkl

∫Ω





1 a2b∗




∫Ω

d2r(s∗ : T(ES)


∫Ω

d2r e∗1 · [∆εr (S)e2]


b =∑ijkl

∫Ω





1 a2b∗




∫Ω

d2r(s∗ : T(ES)


∫Ω

d2r e∗1 · [∆εr (S)e2]


b =∑ijkl

∫Ω



Optical power and acoustic energy

Norm of numerically computed eigenmodes is arbitrary.

Modes must be normalized with respect to powers or energies.

In our gain expression this is explicitly included.

Power normalization factor of optical eigenmode:

P1 =P2 = 2

∫Ω

d2r (e∗xhy − e∗yhx).︸︷︷︸in COMSOL: 2 × emw.Poavz

Energy normalization factor of acoustic eigenmode:

Eb =2

∫Ω

d2r ρ|∂tu|2.







P1 =P2 = 2

∫Ω



Eb =2

∫Ω

d2r ρ|∂tu|2.







P1 =P2 = 2

∫Ω



Eb =2

∫Ω

d2r ρ|∂tu|2.







P1 =P2 = 2

∫Ω



Eb =2

∫Ω

d2r ρ|∂tu|2.







P1 =P2 = 2

∫Ω



Eb =2

∫Ω

d2r ρ|∂tu|2.


Electrostrictive SBS gain summary

From these expressions involving optical and acoustic eigenmodes:

Q =∑ijkl

∫Ω

d2r [e2]∗i [e1]jpijklskl , Eb =2

∫Ω

d2r ρ|∂tu|2,

P1 =P2 = 2

∫Ω

d2r (e∗xhy − e∗yhx),︸︷︷︸in COMSOL: 2 × emw.Poavz

QF =1000.

you can compute the SBS power gain


Your turn: Implement integrals and compute SBS-gain




Q =∑ijkl

∫Ω


∫Ω

d2r ρ|∂tu|2,

P1 =P2 = 2

∫Ω


QF =1000.







Q =∑ijkl

∫Ω


∫Ω

d2r ρ|∂tu|2,

P1 =P2 = 2

∫Ω


QF =1000.





Workshop: Opto-acoustics with COMSOL - CUDOScudos.org.au/nusod_tutorial.pdf · Workshop: Opto-acoustics with COMSOL Christian Wol 1;3 and Michael J. A. Smith 2 1: Centre for Ultrahigh

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