workshop: opto-acoustics with comsol - cudoscudos.org.au/nusod_tutorial.pdf · workshop:...

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

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.

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

sound wave transmittance

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.

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:

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:

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

shear strain

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

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)

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

point mass:

center of mass:

solid body:

point mass:

center of mass:

solid body:

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:

S1 S6 S5

S6 S2 S4

S5 S4 S3

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

S1 S6 S5

S6 S2 S4

S5 S4 S3

T1 T6 T5

T6 T2 T4

T5 T4 T3

Voigt notation

S1 S6 S5

S6 S2 S4

S5 S4 S3

T1 T6 T5

T6 T2 T4

T5 T4 T3

Voigt notation

S1 S6 S5

S6 S2 S4

S5 S4 S3

T1 T6 T5

T6 T2 T4

T5 T4 T3

Voigt notation

S1 S6 S5

S6 S2 S4

S5 S4 S3

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

(∂uj∂xi

+∂ui∂xj

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

(∂uj∂xi

+∂ui∂xj

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

(∂uj∂xi

+∂ui∂xj

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

(∂uj∂xi

+∂ui∂xj

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

(∂uj∂xi

+∂ui∂xj

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)

Photoelastic effect

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

Photoelastic effect

Simple picture:

Higher P-field

Photoelastic effect

Simple picture:

Higher P-field

Photoelastic effect

Simple picture:

Higher P-field

Photoelastic effect

Simple picture:

Higher P-field

Photoelastic effect

Simple picture:

Higher P-field

Photoelastic effect

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)〉 =

d2r(s∗ : T(ES)

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

d2r e∗1 · [∆εr (S)e2]

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

b =∑ijkl

d2r [e2]∗i [e1]jpijklskl

1 a2b∗

d2r(s∗ : T(ES)

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

d2r e∗1 · [∆εr (S)e2]

b =∑ijkl

1 a2b∗

d2r(s∗ : T(ES)

d2r e∗1 · [∆εr (S)e2]

b =∑ijkl

1 a2b∗

d2r(s∗ : T(ES)

d2r e∗1 · [∆εr (S)e2]

b =∑ijkl

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

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

Energy normalization factor of acoustic eigenmode:

d2r ρ|∂tu|2.

P1 =P2 = 2

d2r ρ|∂tu|2.

P1 =P2 = 2

d2r ρ|∂tu|2.

P1 =P2 = 2

d2r ρ|∂tu|2.

P1 =P2 = 2

d2r ρ|∂tu|2.

Electrostrictive SBS gain summary

From these expressions involving optical and acoustic eigenmodes:

Q =∑ijkl

d2r [e2]∗i [e1]jpijklskl , Eb =2

d2r ρ|∂tu|2,

P1 =P2 = 2

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

d2r ρ|∂tu|2,

P1 =P2 = 2

QF =1000.

Q =∑ijkl

d2r ρ|∂tu|2,

P1 =P2 = 2

QF =1000.

workshop: opto-acoustics with comsol - cudoscudos.org.au/nusod_tutorial.pdf · workshop:...

Documents