Phononic CrystalsJ.H. Page

University of ManitobawithSuxia Yang and M.L. Cowan at U of M, Ping Sheng and C.T. Chan at HKUST, &Zhengyou Liu at Wuhan University.

We study ultrasonic waves in complex (strongly scattering) media.

Ordered media (phononic crystals):- ultrasound tunneling- focusing effects

Random media: - ballistic and diffusive wave transport- new ultrasound scattering techniques (DSS & DAWS)

For more info and papers, see www.physics.umanitoba.ca/~jhpage

Outline: Phononic Crystals

Motivation: Why study phononic crystals? (acoustic and elastic analoguesof photonic crystals):

Possible applications- lenses and filters?- sound insulation?

Conclusions

Wave phenomena in 3D phononic crystals (experiments and theory) : - our crystals and the experimental setup- spectral gaps - ultrasound tunneling- near field imaging- focusing effects

-4

-2

0

2

4-4

-2

0

2

4

y (cm

)

x (cm)

PHONONIC CRYSTALSe.g. ordered arrays of mm-sized beads in a liquid or solid matrix �acoustic and elastic analogues of photonic crystals.

Why study phononic crystals?• richer physics?

•scattering contrast � � may be easier to achievecomplete spectral gaps•longitudinal + transverse modes � novel wave phenomena?

• ultrasonic techniques have some advantages•measure the field not the intensity•pulsed techniques are easy

• good theory is available - e.g. Multiple Scattering Theory for elastic andacoustic waves. [Kafesaki and Economou, PRB 60, 11993 (1999); Liu et al. PRB 62,2446 (2000); Psarobas et al., Phys. Rev. B 62, 278 (2000)]

• relatively few experiments have been performed on 3D systems (unlike2D).•new applications?

�� �v,

Questions:

• Can complete spectral gaps be readily achieved?

• How do waves travel through phononic band gaps? No propagating mode � Tunneling?

How long does it take?

• Can phononic crystals be used to focus ultrasound?

Our 3D Phononic Crystals:

Close-packed periodic arrays ofspherical beads surrounded by a liquidor solid matrix:

• hcp arrays of stainless steel balls inwater

• fcc arrays of tungsten carbide balls inwater

• fcc arrays of tungsten carbide beadsin epoxy

For all these crystals:

• very high contrast scatterers Zball/Zmatrix � 30 to 60

• very monodisperse spheres: diameter d = 0.800 � 0.0006 mm

• very high quality crystals: produced by a manual assembly techniqueusing a hexagonal template.

for hcp: ABAB… layer sequence. (slabs with c-axis � layers)

for fcc: ABCABC… layer sequence. ( [111]-axis � layers)

Schematic graph shows how the beads are packed in a FCC crystal along the [111] direction.Beads A are on the bottom layer, B the second and C the third. The sequence is ABCABC...

A

B B B B

B B BC

C

C C

C C

C

Matrix:

water

Lv = 1.49 km/s

� = 1.0 kg/m3

Scatterers:0.800-mm-diametertungsten carbide beads

LvTv

= 6.65 km/s= 3.23 km/s

� = 13.8 kg/m3

Material parameters forour phononic crystalsmade from tungstencarbide beads in water

Sample

SampleHolder

Ultrasound GeneratingTransducer

Ultrasound DetectingTransducer

Close-up of sample

y1

y2L

phase velocity:vp(�) = � L/(� + 2�n)(� = phase delay)

group velocity: vg(�) = L/tpeak

(tpeak = peak delay time)

transmissioncoefficient: Atrans(�) / Aref(FFT ratio)

Experimental setup

Planar input pulse(far field; large y1)

Compare transmittedpulses with referencepulses to measure:

Multiple Scattering TheoryMultiple scattering theory (MST) for acoustic and elastic waves -- ideally suitedto spherical scatterers. (cf. KKR theory) [Liu et al. PRB 62, 2446 (2000) ]

Band-structure: calculate elastic Mie scattering of waves from all the scatterers inthe crystal, and solve the resulting secular equation for the eigenfrequencies.

Transmission:calculate thetransmission througha multi-layer samplewith thickness Lusing a layer MST.

2(2)IQ

(2)IVQ

(2)IIIQ

(2)IIQ

1(1)IQ

(1)IVQ

(1)IIIQ

(1)IIQ

12(12)IQ

(12)IVQ

(12)IIIQ

(12)IIQ

( , ) ( , ) exp[ ( , )]T L A L i L� � � ��

A normalized amplitude (transmission coefficient) � cumulative phase relative to the incident wave.

Transmitted wave

( ) /

( )

p

g

L

d dLdk d

� � �

� ��

�

�

� �

v

v

Phase velocity

Group velocity

�

0.0

0.2

0.4

0.6

0.8

1.0

KHLA�MK

� a

/ 2�

vw

hcp bandstructure: stainless steel spheres in water

a lattice constantsound velocity in water.wv

WAVEVECTOR

Very small complete gap

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

WAVE VECTORKWX�LUX

� a

/2�

vw

Fcc bandstructure: tunsgsten carbide beads in water

a lattice constantsound velocity in water.wv Reciprocal

latticeComplete gap: centre 19 %� �� �

Fcc bandstructure and transmission: tunsgsten carbide beads in epoxy

An even biggercomplete gap!

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

WAVE VECTORUXX U L �

FREQ

UEN

CY (M

Hz)

0.1 0.011E-31E-41E-51E-61E-71E-80.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.57-Layer Sample

K

TRANSMISSION COEFFICIENT

centre 90 %� �� �

0-1.0

-0.5

0.0

0.5

1.0

NO

RMA

LIZE

D F

IELD

0 10 20 30 40 50

-0.08

-0.04

0.00

0.04

0.086-layer sample

TRANSMITTED PULSEINPUT PULSE

TIME (�s)

FCC tungsten carbide beads in water:Input and transmitted pulses near the lowest band gap, k [111]�

0.8 1.20.0

0.2

0.4

0.6

0.8

1.0 INPUT

NO

RMA

LIZE

D A

MPL

ITU

DE

FREQUENCY (MHz)0.8 1.2

0.00

0.04

0.08

0.12

0.16TRANSMITTED

FCC tungsten carbide beads in water:Fourier Spectrum of these input and transmitted pulses near the

lowest band gap0

-1.0

-0.5

0.0

0.5

1.0

NO

RMA

LIZE

D F

IELD

0 10 20 30 40 50

-0.08

-0.04

0.00

0.04

0.086-layer sample

TRANSMITTED PULSEINPUT PULSE

TIME (�s)

0.6 0.8 1.0 1.2 1.4

1E-3

0.01

0.1

1

6 LAYER: DATA THEORY

12 LAYER: DATA THEORY

AM

PLIT

UD

E TR

AN

SMIS

SIO

N

COEF

FICI

ENT

FREQUENCY (MHz)

Transmission coefficient

0.2 0.4 0.6 0.8 1.0 1.2 1.41.0

1.2

1.4

1.6 EXPERIMENT THEORY

PHA

SE V

ELO

CIT

Y (k

m/s

)

FREQUENCY (MHz)

0

20

40

60 EXPERIMENT THEORY

CU

MU

LATI

VE

PHA

SE (r

ad)

Cumulative Phaseand the

Phase Velocity

Use these data tocompute thedispersion curve,� versus k = � /vp,and compare withthe bandstructurecalculation…

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

WAVE VECTORKWX�LUX

� a

/2�

vw

Fcc bandstructure: tunsgsten carbide beads in waterThe black symbols show experimental data from phase velocity measurements

a lattice constantsound velocity in water.wv Reciprocal

latticeComplete gap: centre 19 %� �� �

-30 -20 -10 0 10 20 30-1

0

1

x1796

THROUGH SAMPLE

TIME (�s)

-1

0

1

x1

THROUGH WATER

NO

RM

ALIZ

ED F

IELD -1

0

1 INPUT

Measuring the group velocity: Digitally filtered input andtransmitted pulses (bandwidth 0.05 MHz) for a 12-layer phononic crystalin the middle of the gap, compared with the pulse transmitted through thesame thickness of water.

Frequency dependence of the group velocity for a 5-layer sample.

0.6 0.8 1.0 1.2 1.40

2

4

6

8 THEORY FIT EXPERIMENT

GR

OU

P VE

LOC

ITY

(km

/s)

FREQUENCY (MHz)

0 2 4 6 80

4

8

12

16

1D Theory

vwater = 1.49

Group Velocity Theory Data

vbead = 6.655

VEL

OCI

TY (k

m/s)

SAMPLE THICKNESS (mm)

Group Velocity versus sample thickness in the middle of the gap

The group velocity increases linearly with thickness! � tunneling.Yang et al., Phys. Rev. Lett. 88, 104301 (2002)

Effect of absorption - the two-modes modelSimple physical picture:

Absorption cuts off the long scattering paths � destructive interference isincomplete.

Effectively, absorption introduces a small additional component having a realwave vector inside the gap.

Two-modes model: (Dominant) Tunneling mode (constant tunneling time) + Propagation mode (constant velocity)

� Group velocity becomes

222

2 2 2 22 2

1

1 1

LLll ba

L L L Ll l l la b a b

ga eae

aae a e ae a e b

LLt

��

� � � �

�

� � � �

�

�

v

vwhere L is the thickness, a is the coupling coefficient, ta and la are thetunneling time and tunneling length, vb is the group velocity of the propagatingmode and lb is its decay length.

0 2 4 6 80

4

8

12

16

1D Theory

vwater

Group Velocity Theory Data

vbead=6.655

VEL

OCI

TY (k

m/s)

SAMPLE THICKNESS (mm)

The effect of absorption: Fits of the two-modes model to 1 D theory and toour data for 3 D phononic crystals (dashed curves).

0 2 4 6 80

4

8

12

16

Phase Velocity Theory Data

1D Theory

vwater

Group Velocity Theory Data

vbead=6.655

VEL

OCI

TY (k

m/s)

SAMPLE THICKNESS (mm)

Add: the phase velocity in the middle of the gap for the same sample thicknesses.

Transmitted waveIncident

wave

V0

Electron tunneling through a barrier

0

0 00

0

xV V x L

x L

���

� � ��� ��

2 2

2 ( ) ( )2

d V E xm dx

��� � � �

�

From the transmitted phase at large large L, find

� �

� �0

0

2arctan

2p

VL

V�

�

� �

� ��� ��

�� �� �

��

� �

v 0g

VL � �� �

� �� �� �

��

v

Barrier: Both phase and group velocities increase with L.

Band gap: Group velocity increases while the phase velocity is constant.

Schrodinger equ.

Boundary conditions

(0 ) (0 ); (0 ) (0 )

( ) ( ); ( ) ( )L L L L

� � � �

� � � �

� � � �

� � � �

� �� �

� �� �

10

1

86421.5

Experiment

Theory

Phononic crystals

Tungsten carbide

water

TRA

NSI

T TI

ME

(�s)

SAMPLE THICKNESS (mm)

Magnitude of the tunneling time: ttunnel ~ 1 / ��gap

Tunneling Time

(also true for lightand electrons!)

Question: Is ultrasound tunneling dispersionless?

Some general points about pulse propagation in dispersive media:

� � � � � �o0, expx t e t i t� �� �

� � � � � �1o, exp exp

2 e

xx t E ik xl

� � � ��

� �� �� �� � � � �� �� � �

� �

� � � � � �2

21o o o2! 2

o o

dk d kk kd d

� � � � �

� �

� � � � � ��

Incident pulse at x = 0 � Fourier transform:

pulse envelope E(�) is F.T. of e(t)

Pulse after traveling a distance x

extinction length, 1 1 1e s al l l� � �

� �

scattering absorption

Expand the wave vector k(�) in a Taylor series about the central frequency�o:

group velocity dispersion (GVD)

� � � � � �ot E� � � �� �� � � � �� �

inverse group velocity

� � � �

� � � � � �� � � � � �� �

� � � � � � � �� � � �� �

o

21 2 21o o o2o o

1 2 2 21o o 2o o

o

, exp exp2

exp exp

exp exp exp exp exp2

exp exp2

e

e

e p g

xx t ik xl

E i dk d x i d k d x

x ik x i t E i dk d x i d k d xl

x x xi t tl

�

� � � � � � � �

� � � � � �

� �

�

�

� �� � � �� �

�

�� � � � � � �� �

� � �� � � � � �� � �� ��

�� � �� � � �� � � � �� � � �

� �� �� �� �

�

v v� � � �� �1 2 2 21

2 oexpe t i d k d x� ��

� ��� � �� �� �� �� �� � �� �� ���

Pulse at (x,t) becomes

attenuation

carrier frequency oscillationstravel at vp = �o/ko

peak travels at the group velocityof the central frequency

GVD changes the width symmetrically

convolution

Good approximation, so long as the pulse bandwidth is sufficiently narrow that:• higher order terms in k(�) expansion are negligible• frequency dependence of le is not important.

0.7 0.8 0.9 1.0 1.1 1.2 1.30

5

10

f (MHz)

0

2

Data Theory

-20

0

20

d2 k/d�

2dk

/d�

k

Frequency dependence of the wave vector, the inverse group velocity and the groupvelocity dispersion near the band gap in a 5 layer tungsten carbide/water crystal.

[c.f. measurements near the band edge in photonic crystals by Imhof et al. PRL 83, 2942 (1999)]

-20 -10 0 10 20 300

1 Shifted Reference Sample6.14

f = 0.744 MHz

TIME (�s)

0

1 Shifted Reference Sample

f = 0.972 MHz1.0

NO

RMA

LIZE

D A

MPL

ITU

DE

0

1 Shifted Reference Sample

f = 1.2 MHz4.9

Pulse broadening (a) just above the band gap, (b) in the middle of the gap and(c) just below the band gap (bandwidth 0.02 MHz). � Very weak dispersionin the gap, despite very strong scattering and a large variation in vp with �.