3/10/06 1

Negative Refraction in 2-d Sonic Crystals

Lance Simms

3/10/06

3/10/06 2

Inspiration

Trying to Sleep on a warm summer Night. Neighbors blasting music. Window shut= no breeze coming through

Window open= music even louder

Neighbor’s speaker What will let breeze through, but not sound?

Possible Answer: Sonic Crystal that blocks 20 to 20000Hz

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Applications

• Sonic Wave Guides– Used to guide acoustic waves– High Transmission/low leakage

for certain frequencies

Figures: Sonic Crystals and Sonic Wave Guides, Miyashita 2002

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Applications

• Inhibition of vibrations– Acoustic: Sound “Barrier”– Mechanical: block elastic waves ar

r=10.2mm

a=24.0mm

Sonic Crystal of acrylic Resin rods in air

Figures: Miyashita, Sonic Crystals and Sonic Wave Guides

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Applications• Acoustic Lensing

– Far-field acoustic imaging– Useful for focusing ultrasound for medical applications

Fascinating Example: Extra- corporeal Shock Wave Osteotomy

The goal of this method is to cut bones in a living body without incising the skin by using high intensity energy of ultrasound and high pressure generated by cavitation (Ukai 2003)Requires < 1mm displacement of focal point

Figure: Ukai, Ultrasound Propagation, 2003

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Definition of Phononic Crystal

Phononic CrystalA periodic array composed of scatterers embedded in a host material

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μh,λ h,Kh ,ρ h

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μs,λ s,Ks,ρ s

host

scatterer

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μ , - Lamé Coefficients

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λ

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ρ -Density

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c longitudinal = K /ρ

-Bulk Modulus

“Unit Cell”Sound Velocities in materials

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K

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c transverse = μ /ρ h-host solid or fluids-scatterer solid or fluid

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And a Sonic Crystal?

Sonic CrystalPhononic crystal that is considered to be indepenedent of shear waves

Scatterers may be solids in fluid host

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λh ,Kh,ρ h

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μs,λ s,Ks,ρ s

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μh = 0

Ignore shear waves in scatterer with large since longitudinal waves in host do not couple with transverse modes in scatterer.

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ρs /ρ hh-host fluids-scatterer solid or fluid

In 2-d:

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f = As / Ac

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As =

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Ac =

area of scatterer

area of unit cell

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The “First” Sonic Crystal

Physics World, Dec. 2005

Sculpture by Eusebrio Sempere exhibited at the Juan March Foundation in Madrid in 1995

2d Square Lattice of steel cylinders in air

arr=1.45 cma=10.0 cmf=0.066

Meseguer et al. measured “pseudo” band-gap at 1.67kHz i.e strong attenuation of sound

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ρh =1.29kg /m3

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ρs = 7670kg /m3

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ch = 340m /s

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cs = 6010m /s

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Explanation of Attenuation

2-d periodic structure

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λ(r r ) = λ (

v r +

r R n )

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ρ(r r ) = ρ (

v r +

r R n )

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ˆ x

€

ˆ y

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rR n = a(nx

ˆ x + nyˆ y )

€

a

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nx,y = −2,−1,0,1,2...

Real Space

Reciprocal Space

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rG n =

2π

a(n1

ˆ x + n2ˆ y )

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n1,2 = −2,−1,0,1,2...

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Γ

€

X€

MIrreducible triangle of First Brillouin Zone (BZ)

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2π /a

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Explanation of Attenuation

Bragg Diffraction

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rk

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Δ r

k

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rk '

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Δ r

k =r G

In 1-d occurs at:

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k = ±nπ /a

So along axis, first diffraction occurs roughly at:

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ΓX

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k =ω

c~

π

a

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rk -incident wavevector

-scattered wavevector

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rk '

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f =c

2a=

340m /s

2(.1)m~ 1.7kHz

n=1,2,3…

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How it began

Next slide

Paper by Kushwaha in 1993 predicts band gaps in elastic periodic media

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Differential Equations

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(λ + 2μ)∇(∇ •r u ) − μ∇ × (∇ ×

r u ) + ρω2 r

u = 0

In Phononic Crystals: Elastc Wave Equation

Inhomogeneous media-transverse and longitudinal components not separable

In Sonic Crystals: Acoustic Wave Equation

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λ∇ ∇p

ρ

⎡

⎣ ⎢

⎤

⎦ ⎥− ρω2 p = 0

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p = −λ∇r u pressure

Coefficients depend on postion and are periodic with period of the crystal

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Methods to predict crystal properties

1) Plane Wave Method (PW)

Expand periodic coefficients in acoustic wave equation as Fourier Series.

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λ(r r ) = λ r

G e i(

r G •

r r )

v G

∑

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ρ(r r ) = ρ r

G e i(

r G •

r r )

v G

∑

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p(r r ) = e i(

r k •

r r ) λ r

k +r

G e i(

r G •

r r )

v G

∑

Use Floquet-Bloch theorem to express pressure field solution as a plane wave modulated by a periodic function.

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Plane Wave Method

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ρ rk +

r G

−1 (r k +

r G )(

r k +

r G ') + λ r

k +v

G

−1 ω2[ ] p r

k +r

G = 0

v G '

∑

Inserting Fourier series expansions in differential equation, assuming harmonic time dependence

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e iωt

For M terms kept in the sum, this is an MxM matrix.Eigenvalues are

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ωn (v k ) n=1 (first band), 2 (second band) …

Scanning Brillouin zone yields

-Dispersion Relation-Equifrequency Surfaces (EFS)

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Γ

€

X€

M

BZ

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PW Results for sculpture

No Complete Band found

Using M=10

Density of states has minima at 1.7 and 2.4kHz

Figure: Kuswaha 1997

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PW Prediction for Steel-AirStrong Band Gaps at1.6-2.4kHz6.7-6.8kHz

What happens here?

f=0.55

f=0.3

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PW Method

For f > .8 Eigenvalues are imaginary

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p(r r ) = e i(

r k •

r r ) λ r

k +r

G e i(

r G •

r r )e−i Re(ω )teIm(ω )t

v G

∑

Solutions would be of the form

Introducing a damping/diverging term

Real Physics? Probably not

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PW Method

Problems and Disadvantages of Plane Wave Method

1) Cannot deal with finite/random media

2) Convergence problems when dealing with systems of very high/very low filling ratios

3) Cannot accommodate transverse modes localized in scatterers (negligible in high density contrast ratio)

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Methods to Predict Crystal Properties

2) Multiple Scattering Method (MS)

Based on Korringa-Kohn-Rostoker’s (KKR) theory from electronic band structure calculations.

For a set of N scatterers located at where i=1,2,…,N the total wave incident on ith scatter is

€

rr i

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pini (

r r ) = po(

r r ) + ps(

r r ,

r r j )

j=1, j≠ i

N

∑

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po(r r ) -source

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ps(v r ,

r r j ) -scattered waves from

all other scatterers€

po scatterers

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rr 1

€

rr 2

€

rr

Allows amplitude of field to be calculated at any point

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Multiple Scattering Method

For Sonic Crystal with N identical Cylinders, scattered wave from jth cylinder is

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ps (r r ,

r r j ) = iπAn

jHn(1)(k

r r −

r r j )e

inφ r r −

r r j

−∞

∞

∑

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Hn(1)

-Hankel function of first kind

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φ rr −

r r j -Azimuthal angle of

relative to x axis

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rr −

r r j

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rr

€

rr j

€

rr −

r r j

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φ rr −

r r j

Total incident wave is given by

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pin (r r ) = iπBn

i Jn(1)(k

r r −

r r i )e

inφ r r −

r r i

−∞

∞

∑Bessel functions Jn are used to ensure p does not diverge atcenter of cylinder

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Multiple Scattering MethodCoefficients Ai

n and Bin are related by boundary conditions

1) Pressure is continuous across interface between cylinder and surrounding medium

2) Normal veloctiy is continuous as well

Defining scattering coefficient that depends on density and contrast ratio

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Γni

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Bni = iπΓn

i Ani

And a structure constant that depends on geometry of scatterers,

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Gl,ni, j

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Γni An

i − Gl,ni, j Al

j = Tni

−n max

n max

∑j, j≠ i

N

∑ Source term

(2n+1)N x (2n+1)N matrix equation

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Multiple Scattering Method

Setting source term to zero, and solving

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det Γni An

i − Gl ,ni, j Al

j

−n max

n max

∑j , j≠ i

N

∑ = 0

-Normal modes are obtained. -For periodic systems, sum over lattice sites yields band structure

Using Coefficients, transmission spectrum can be obtained €

Ani

TheoryExperiment

-Without crystal

-With crystal

Figures: Chen et. al 2003, Robertson 1998

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Difference with Nmax

Nmax=1 Nmax=2

Nmax=4Nmax=3

1GB Memory not enough for Nmax = 5

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MS Method Applied to Water Waves

Experiment

Simulation

MST predicts negative refraction and it is observed!

Figure: X. Hu et al., Superlensing in liquid surface waves(2004)

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Negative Refraction in Sonic Crystals

PR/NR--Positive/Negative Refraction SC/PC--Sonic/Photonic Crystal

2 Types of Negative Refraction 1) Backward

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rS •

r k > 0

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rS •

r k < 0

2) Forward

Figure: Feng, Acoustic Backward-Wave Negative Refraction 2006

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Backward Negative Refraction

Steel Rods immersed in Water

Figure: Sukhovich, Negative Refraction of ultrasonic waves

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Forward Negative Refraction

Forward Negative Refraction can Forward Negative Refraction can occur with occur with

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rV g •

r k > 0

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sgn(r

V g •r k ) = sgn(

r S •

r k )

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rS •

r k > 0

Brillouin zones of model Photonic Crystal In regions of negative phononic effective mass

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∂2ω /∂ki∂k j < 0Regions around M point have

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∂2ω /∂ki∂k j < 0 (Green Triangle)

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Simulating PWNRX. Zhang et al. used the MS method to simulate forward negative refraction in

first band of phononic/sonic crystals

All incoming angles negatively refract

They looked for regions of All-Angle Negative Refraction (AANR)

(i) The EFS of the crystal is all convex with a negative phononic effective mass

(ii) All incoming wave vectors at such a frequency are included within constant-frequency contour of crystal

(iii) The frequency is below

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πc /a(below this line)

Figures: X. Zhang et al. (2004)

Results are shown for mercury/water crystals

For Steel Air systems?

Mercury (EFS)

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2-d Steel/Air Sonic CrystalsNo regions of AANR found in steel/air square lattice

Large regions of Forward Negative Refraction

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∂2ω /∂ki∂k j < 0

Figures are for mercury/water. Similar ones were demonstrated for steel/air (Xhang et.al)

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BWNR in Steel-Air systems Further study to find backward negative refraction in second band of 2-d triangular sonic crystals using MS simulations

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rS •

r k < 0

For f=0.47, in second band EFSs move inwards with increasing frequency so

In frequency range of ~.65-.95 EFS are roughly circular

Can define effective refractive index (ERI)

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n = −r k c /ω

Figures: Zhang. et al. (2005)

Use ERI in Snell’s Law, Brewster’s angle etc.

f=0.47

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Using negative ERI for imaging

In order to demonstrate acoustic imaging with negative refraction, MS was applied to the following setup

Point source placed at O

Ray trace diagram used to define

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D2 = (1+ n2 − sin2 α /cosα )D1

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D3 = (1+ cosα / n2 − sin2 α )d

If frequency can be found such that n=-1

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D2 = 2D1

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D3 = 2d

Thickness of slab

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d =3

2(l −1)a + 2r

- # layers- radius of

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l

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rcylinder

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Using negative ERI for imaging

For the EFS at and

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λ =1.54a

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ω =0.65The ERI is

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n ≈ −.7

Brewster’s Angle is given by

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θ ≈sin−1(0.7) ≈ 44°

Near axis approximation gives

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D2 = (1+ n )D1

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D3 = (1+1/ n )d Thickness of slab

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d =3

2(l −1)a + 2r

- # layers- radius of

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l

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rcylinderExpect rays to converge near I’

and I but “out of focus”

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Simulation Results n=-.7

Predicted Simulation result

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D1 = 6aFor

Distance

9 layer sample-- d=7.76a

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D2

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D3

10.2a

18.2a

10.1a*

19.3a

Predicted Simulation result

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D1 = 6aFor

Distance

15 layer sample-- d=12.76a

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D2

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D3

10.2a

31.5a

10.1a*

33.8a*

Predicted

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D1 =11aFor

Distance

15 layer sample-- d=12.76a

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D2

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D3

18.7a

31.5a

18.5a

33.8a

Simulation result

*

*

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D3 Independent of

€

D1

*

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D2 Independent of

€

d

Intensity

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Simulations for n=-1

For f=0.906 EFS at and

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λ =2.74a

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ω =0.365The ERI is

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n ≈ −1 Layers

6

7

9

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D1 = 6a For all images

d=5.33a

D3=10.8a

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D3 ≈ 2d

d=6.20a

D3=12.5a

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D3 ≈ 2d

d=7.93a

D3=16.1a

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D3 ≈ 2dPlots are of pressure:Source in phase with image

D1 fixed----Vary Thickness

3/10/06 35

Simulations for n=-1

For f=0.906 EFS at and

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λ =2.74a

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ω =0.365The ERI is

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n ≈ −1

6 layers for all imagesD1=0.5a

D3=10.8a

D3=10.8a

D3=10.8a

Plots are of pressure:Source in phase with image

D1=2.5a

D1=4.5a

D3 independent of D1

Thickness fixed----Vary D1

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Reproducing results

For n=-.7 pattern is similar, shows focusing effect

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Results and Conclusion

MS methods show that negative refraction and acoustic imaging can occur in 2-dimensional sonic crystals composed of steel cylinders in an air background

Now it is time for experiments to verify this.

One step closer to sleeping with loud neighbors

3/10/06 38

Additional Slides