Phononic Crystals. J.H. Page

Transcription

1 Phononic Crystals J.H. Page University of Manitoba with Suxia 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. Random media: - ballistic and diffusive wave transport - new ultrasound scattering techniques (DSS & DAWS) Ordered media (phononic crystals): - ultrasound tunneling - focusing effects For more info and papers, see

2 Outline: Phononic Crystals Motivation: Why study phononic crystals? (acoustic and elastic analogues of photonic crystals): Wave phenomena in 3D phononic crystals (experiments and theory) : - our crystals and the experimental setup - spectral gaps - ultrasound tunneling - near field imaging - focusing effects Possible applications - lenses and filters? - sound insulation? y (cm) 4 Conclusions x (cm) 2 4-4

3 PHONONIC CRYSTALS e.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, v may be easier to achieve complete 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 and acoustic waves. [Kafesaki and Economou, PRB 6, (1999); Liu et al. PRB 62, 2446 (2); Psarobas et al., Phys. Rev. B 62, 278 (2)] relatively few experiments have been performed on 3D systems (unlike 2D). new applications?

4 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?

5 Our 3D Phononic Crystals: Close-packed periodic arrays of spherical beads surrounded by a liquid or solid matrix: hcp arrays of stainless steel balls in water fcc arrays of tungsten carbide balls in water fcc arrays of tungsten carbide beads in epoxy For all these crystals: very high contrast scatterers Z ball /Z matrix 3 to 6 very monodisperse spheres: diameter d =.8.6 mm very high quality crystals: produced by a manual assembly technique using a hexagonal template.

6 for hcp: ABAB layer sequence. (slabs with c-axis layers) for fcc: ABCABC layer sequence. ( [111]-axis layers) A C C C B B B B C C C B B B C 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... Material parameters for our phononic crystals made from tungsten carbide beads in water Scatterers:.8-mm-diameter tungsten carbide beads = 13.8 kg/m 3 v L v T = 6.65 km/s = 3.23 km/s Matrix: water = 1. kg/m 3 v L = 1.49 km/s

7 Experimental setup Ultrasound Detecting Transducer Compare transmitted pulses with reference pulses to measure: Sample Sample Holder y 2 L Close-up of sample phase velocity: v p () = L/( + 2n) ( = phase delay) group velocity: Planar input pulse (far field; large y 1 ) y 1 Ultrasound Generating Transducer v g () = L/t peak (t peak = peak delay time) transmission coefficient: A trans () / A ref (FFT ratio)

8 Multiple Scattering Theory Multiple scattering theory (MST) for acoustic and elastic waves -- ideally suited to spherical scatterers. (cf. KKR theory) [Liu et al. PRB 62, 2446 (2) ] Band-structure: calculate elastic Mie scattering of waves from all the scatterers in the crystal, and solve the resulting secular equation for the eigenfrequencies. Transmission: calculate the transmission through a multi-layer sample with thickness L using a layer MST. Q Q I IV (1) (1) 1 Q Transmitted wave III II Q (1) (1) I Q IV Q (2) (2) 2 III Q II Q (2) (2) Q Q I IV (12) (12) T( L, ) A( L, )exp[ i( L, )] A normalized amplitude (transmission coefficient) cumulative phase relative to the incident wave. Phase velocity Group velocity v v p g ( ) L / ( ) d dk d L d 12 Q III II Q (12) (12)

9 hcp bandstructure: stainless steel spheres in water 1..8 a / 2 v w K M A L H K a lattice constant sound velocity in water. v w WAVEVECTOR Very small complete gap

10 Fcc bandstructure: tunsgsten carbide beads in water a/2 v w X U a lattice constant v sound velocity in water. w Complete gap: centre 19 % L WAVE VECTOR X Reciprocal lattice W K

11 Fcc bandstructure and transmission: tunsgsten carbide beads in epoxy Layer Sample FREQUENCY (MHz) An even bigger complete gap! centre 9 %.5.5. X U L X WAVE VECTOR U K..1.11E-31E-41E-51E-61E-71E-8 TRANSMISSION COEFFICIENT

12 FCC tungsten carbide beads in water: Input and transmitted pulses near the lowest band gap, k [111] INPUT PULSE 1. TRANSMITTED PULSE 6-layer sample.8 NORMALIZED FIELD TIME (s) -.8

13 FCC tungsten carbide beads in water: Fourier Spectrum of these input and transmitted pulses near the -1. lowest band gap NORMALIZED FIELD INPUT PULSE TRANSMITTED PULSE 6-layer sample TIME (s) -.8 NORMALIZED AMPLITUDE 1. INPUT TRANSMITTED FREQUENCY (MHz)

14 Transmission coefficient AMPLITUDE TRANSMISSION COEFFICIENT E-3 6 LAYER: DATA THEORY 12 LAYER: DATA THEORY FREQUENCY (MHz)

15 Cumulative Phase and the Phase Velocity CUMULATIVE PHASE (rad) EXPERIMENT THEORY Use these data to compute the dispersion curve, versus k = /v p, and compare with the bandstructure calculation PHASE VELOCITY (km/s) EXPERIMENT THEORY FREQUENCY (MHz)

16 Fcc bandstructure: tunsgsten carbide beads in water The black symbols show experimental data from phase velocity measurements a/2 v w X U a lattice constant v sound velocity in water. w Complete gap: centre 19 % L WAVE VECTOR X Reciprocal lattice W K

17 Measuring the group velocity: Digitally filtered input and transmitted pulses (bandwidth.5 MHz) for a 12-layer phononic crystal in the middle of the gap, compared with the pulse transmitted through the same thickness of water. 1 INPUT NORMALIZED FIELD THROUGH WATER x1 THROUGH SAMPLE -1 x TIME (s)

18 Frequency dependence of the group velocity for a 5-layer sample. GROUP VELOCITY (km/s) THEORY FIT EXPERIMENT FREQUENCY (MHz)

19 Group Velocity versus sample thickness in the middle of the gap D Theory Group Velocity Theory Data VELOCITY (km/s) 8 4 v bead = v water = SAMPLE THICKNESS (mm) The group velocity increases linearly with thickness! tunneling. Yang et al., Phys. Rev. Lett. 88, 1431 (22)

20 Effect of absorption - the two-modes model Simple physical picture: Absorption cuts off the long scattering paths destructive interference is incomplete. Effectively, absorption introduces a small additional component having a real wave vector inside the gap. Two-modes model: (Dominant) Tunneling mode (constant tunneling time) + Propagation mode (constant velocity) Group velocity becomes v g L L L 2l 2 2l 1 b ae a a e L L ta L L 2la 2 2lb 2la 2 2lb L v ae 1a e ae 1a e b where L is the thickness, a is the coupling coefficient, t a and l a are the tunneling time and tunneling length, v b is the group velocity of the propagating mode and l b is its decay length.

21 The effect of absorption: Fits of the two-modes model to 1 D theory and to our data for 3 D phononic crystals (dashed curves) D Theory Group Velocity Theory Data VELOCITY (km/s) 8 4 v bead =6.655 v water SAMPLE THICKNESS (mm)

22 Add: the phase velocity in the middle of the gap for the same sample thicknesses D Theory Group Velocity Theory Data VELOCITY (km/s) 8 4 v bead =6.655 v water Phase Velocity Theory Data SAMPLE THICKNESS (mm)

23 Electron tunneling through a barrier V Schrodinger equ. 2 2 d ( V E) ( x) 2 2mdx Boundary conditions ( ) ( ); ( ) ( ) ( L ) ( L ); ( L ) ( L ) Incident wave Transmitted wave x V V x L x L From the transmitted phase at large large L, find arctan 2 2 V v p L vg L V V Barrier: Both phase and group velocities increase with L. Band gap: Group velocity increases while the phase velocity is constant.

24 Tunneling Time water TRANSIT TIME (s) 1 Phononic crystals Experiment Theory Tungsten carbide SAMPLE THICKNESS (mm) 8 1 Magnitude of the tunneling time: t tunnel ~ 1 / gap (also true for light and electrons!)

25 Question: Is ultrasound tunneling dispersionless? Some general points about pulse propagation in dispersive media: Incident pulse at x = Fourier transform: x, t etexpi t Pulse after traveling a distance x, 1 x x t exp Eo expikx 2l e k 1 k o pulse envelope E() is F.T. of e(t) extinction length, dk d k 2 o o 2! 2 o d o d o 2 t E e s a l l l scattering absorption Expand the wave vector k() in a Taylor series about the central frequency o : o inverse group velocity group velocity dispersion (GVD)

26 Pulse at (x,t) becomes x o 2le E o exp i dk d o x exp i 2 d k d o x o o xt, exp exp ikx 1 o x exp exp ikox exp iot E exp i dk d x exp i d k d x 2 2l o e x x x exp exp io t t et exp i dkd x 2 2le vp v o g convolution attenuation peak travels at the group velocity of the central frequency carrier frequency oscillations travel at v p = o /k o GVD changes the width symmetrically Good approximation, so long as the pulse bandwidth is sufficiently narrow that: higher order terms in k() expansion are negligible frequency dependence of l e is not important.

27 Frequency dependence of the wave vector, the inverse group velocity and the group velocity 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)] 2 d 2 k/d 2 dk/d -2 2 Data Theory 1 5 k f (MHz)

28 Pulse broadening (a) just above the band gap, (b) in the middle of the gap and (c) just below the band gap (bandwidth.2 MHz). Very weak dispersion in the gap, despite very strong scattering and a large variation in v p with f = 1.2 MHz Shifted Reference Sample NORMALIZED AMPLITUDE 1 1. f =.972 MHz Shifted Reference Sample 1 f =.744 MHz Shifted Reference 6.14 Sample TIME (s)