Negative Refraction in 2 d Sonic Crystals Lance

Negative Refraction in 2 -d Sonic Crystals Lance Simms 3/10/06 1

Inspiration Trying to Sleep on a warm summer Night. Neighbors blasting music. Window shut= no breeze coming through Window open= music even louder Possible Answer: Sonic Crystal that blocks 20 to 20000 Hz Neighbor’s speaker 3/10/06 What will let breeze through, but not sound? 2

Applications • Sonic Wave Guides – Used to guide acoustic waves – High Transmission/low leakage for certain frequencies 3/10/06 Figures: Sonic Crystals and Sonic Wave Guides, Miyashita 2002 3

Applications • Inhibition of vibrations – Acoustic: Sound “Barrier” – Mechanical: block elastic waves r=10. 2 mm a=24. 0 mm r a Sonic Crystal of acrylic Resin rods in air 3/10/06 Figures: Miyashita, Sonic Crystals and Sonic Wave Guides 4

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 < 1 mm displacement of focal point 3/10/06 Figure: Ukai, Ultrasound Propagation, 2003 5

Definition of Phononic Crystal host A periodic array composed of scatterers embedded in a host material , - Lamé Coefficients scatterer -Density -Bulk Modulus Sound Velocities in materials “Unit Cell” h-host s-scatterer 3/10/06 solid or fluid 6

And a Sonic Crystal? Sonic Crystal Phononic crystal that is considered to be indepenedent of shear waves area of scatterer Scatterers may be solids in fluid host Ignore shear waves in scatterer with large since longitudinal waves in host do not couple with transverse modes in scatterer. 3/10/06 area of unit cell h-host s-scatterer fluid solid or fluid In 2 -d: 7

The “First” Sonic Crystal Sculpture by Eusebrio Sempere exhibited at the Juan March Foundation in Madrid in 1995 2 d Square Lattice of steel cylinders in air r=1. 45 cm a=10. 0 cm f=0. 066 r a Meseguer et al. measured “pseudo” band-gap at 1. 67 k. Hz i. e strong attenuation of sound 3/10/06 Physics World, Dec. 2005 8

Explanation of Attenuation 2 -d periodic structure Real Space Reciprocal Space Irreducible triangle of First Brillouin Zone (BZ) 3/10/06 9

Explanation of Attenuation Bragg Diffraction In 1 -d occurs at: -incident wavevector -scattered wavevector So along 3/10/06 n=1, 2, 3… axis, first diffraction occurs roughly at: 10

How it began Next slide Paper by Kushwaha in 1993 predicts band gaps in elastic periodic media 3/10/06 11

Differential Equations In Phononic Crystals: Elastc Wave Equation Inhomogeneous media-transverse and longitudinal components not separable In Sonic Crystals: Acoustic Wave Equation pressure Coefficients depend on postion and are periodic with period of the crystal 3/10/06 12

Methods to predict crystal properties 1) Plane Wave Method (PW) Expand periodic coefficients in acoustic wave equation as Fourier Series. Use Floquet-Bloch theorem to expressure field solution as a plane wave modulated by a periodic function. 3/10/06 13

Plane Wave Method Inserting Fourier series expansions in differential equation, assuming harmonic time dependence For M terms kept in the sum, this is an Mx. M matrix. Eigenvalues are n=1 (first band), 2 (second band) … Scanning Brillouin zone yields -Dispersion Relation -Equifrequency Surfaces (EFS) 3/10/06 BZ 14

PW Results for sculpture Using M=10 No Complete Band found Density of states has minima at 1. 7 and 2. 4 k. Hz 3/10/06 Figure: Kuswaha 1997 15

PW Prediction for Steel-Air Strong Band Gaps at 1. 6 -2. 4 k. Hz 6. 7 -6. 8 k. Hz What happens here? f=0. 55 f=0. 3 3/10/06 16

PW Method For f >. 8 Eigenvalues are imaginary Solutions would be of the form Introducing a damping/diverging term Real Physics? Probably not 3/10/06 17

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) 3/10/06 18

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 scatterers -source -scattered waves from all other scatterers 3/10/06 Allows amplitude of field to be calculated at any point 19

Multiple Scattering Method For Sonic Crystal with N identical Cylinders, scattered wave from jth cylinder is -Hankel function of first kind -Azimuthal angle of relative to x axis Total incident wave is given by 3/10/06 Bessel functions Jn are used to ensure p does not diverge at center of cylinder 20

Multiple Scattering Method Coefficients Ain 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 and contrast ratio And a structure constant scatterers, that depends on density that depends on geometry of Source term (2 n+1)N x (2 n+1)N matrix equation 3/10/06 21

Multiple Scattering Method Setting source term to zero, and solving -Normal modes are obtained. -For periodic systems, sum over lattice sites yields band structure Using Coefficients, transmission spectrum can be obtained Experiment Theory -Without crystal -With crystal 3/10/06 22 Figures: Chen et. al 2003, Robertson 1998

Difference with Nmax 3/10/06 Nmax=1 Nmax=2 Nmax=3 Nmax=4 1 GB Memory not enough for Nmax = 5 23

MS Method Applied to Water Waves Experiment Simulation Figure: X. Hu et al. , Superlensing in liquid surface waves(2004) MST predicts negative refraction and it is observed! 3/10/06 24

Negative Refraction in Sonic Crystals PR/NR--Positive/Negative Refraction SC/PC--Sonic/Photonic 2 Types of Negative Refraction 1) Backward 2) Forward 3/10/06 Figure: Feng, Acoustic Backward-Wave Negative Refraction 2006 Crystal 25

Backward Negative Refraction Steel Rods immersed in Water 3/10/06 Figure: Sukhovich, Negative Refraction of ultrasonic waves 26

Forward Negative Refraction can occur with In regions of negative phononic effective mass Brillouin zones of model Photonic Crystal Regions around M point have (Green Triangle) 3/10/06 27

Simulating PWNR X. Zhang et al. used the MS method to simulate forward negative refraction in first band of phononic/sonic crystals Mercury (EFS) 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 constantfrequency contour of crystal (iii) The frequency is below (below this line) All incoming angles negatively refract Results are shown for mercury/water crystals For Steel Air systems? 3/10/06 Figures: X. Zhang et al. (2004) 28

2 -d Steel/Air Sonic Crystals No regions of AANR found in steel/air square lattice Large regions of Forward Negative Refraction Figures are for mercury/water. Similar ones were demonstrated for steel/air (Xhang et. al) 3/10/06 29

BWNR in Steel-Air systems Further study to find backward negative refraction in second band of 2 -d triangular sonic crystals using MS simulations 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) Use ERI in Snell’s Law, Brewster’s angle etc. 3/10/06 Figures: Zhang. et al. (2005) f=0. 47 30

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 If frequency can be found such that n=-1 Thickness of slab - # layers - radius of cylinder 3/10/06 31

Using negative ERI for imaging For the EFS at The ERI is and Brewster’s Angle is given by Near axis approximation gives Thickness of slab Expect rays to converge near I’ and I but “out of focus” 3/10/06 - # layers - radius of cylinder 32

Simulation Results n=-. 7 Intensity For 9 layer sample-- d=7. 76 a Distance Predicted 10. 2 a 18. 2 a For 15 layer sample-- d=12. 76 a Simulation result Predicted 10. 2 a 10. 1 a* 33. 8 a* 31. 5 a Distance For 15 layer sample-- d=12. 76 a Distance 3/10/06 * * Simulation result 10. 1 a* 19. 3 a Predicted 18. 7 a 31. 5 a Independent of Simulation result 18. 5 a 33. 8 a * 33

Simulations for n=-1 For f=0. 906 EFS at The ERI is For all images and d=5. 33 a Layers D 3=10. 8 a 6 d=6. 20 a D 3=12. 5 a 7 d=7. 93 a D 3=16. 1 a 9 Plots are of pressure: Source in phase with image 3/10/06 D 1 fixed----Vary Thickness 34

Simulations for n=-1 For f=0. 906 EFS at The ERI is 6 layers for all images and Thickness fixed----Vary D 1=0. 5 a D 3=10. 8 a D 1=2. 5 a D 3=10. 8 a D 1=4. 5 a D 3=10. 8 a Plots are of pressure: Source in phase with image 3/10/06 D 3 independent of D 1 35

Reproducing results For n=-. 7 pattern is similar, shows focusing effect 3/10/06 36

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 37

Additional Slides 3/10/06 38