3/10/061 Negative Refraction in 2-d Sonic Crystals Lance Simms 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

3/10/063 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

3/10/064 Applications Inhibition of vibrations –Acoustic: Sound “Barrier” –Mechanical: block elastic waves a r r=10.2mm a=24.0mm Sonic Crystal of acrylic Resin rods in air Figures: Miyashita, Sonic Crystals and Sonic Wave Guides

3/10/065 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

3/10/066 Definition of Phononic Crystal Phononic Crystal A periodic array composed of scatterers embedded in a host material host scatterer, - Lamé Coefficients -Density -Bulk Modulus “Unit Cell” Sound Velocities in materials h-host solid or fluid s-scatterer solid or fluid

3/10/067 And a Sonic Crystal? Sonic Crystal Phononic crystal that is considered to be indepenedent of shear waves 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. h-host fluid s-scatterer solid or fluid In 2-d: area of scatterer area of unit cell

3/10/068 The “First” Sonic Crystal Physics World, Dec Sculpture by Eusebrio Sempere exhibited at the Juan March Foundation in Madrid in d Square Lattice of steel cylinders in air a r r=1.45 cm a=10.0 cm f=0.066 Meseguer et al. measured “pseudo” band-gap at 1.67kHz i.e strong attenuation of sound

3/10/069 Explanation of Attenuation 2-d periodic structure Real Space Reciprocal Space Irreducible triangle of First Brillouin Zone (BZ)

3/10/0610 Explanation of Attenuation Bragg Diffraction In 1-d occurs at: So along axis, first diffraction occurs roughly at: -incident wavevector -scattered wavevector n=1,2,3…

3/10/0611 How it began Next slide Paper by Kushwaha in 1993 predicts band gaps in elastic periodic media

3/10/0612 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

3/10/0613 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 express pressure field solution as a plane wave modulated by a periodic function.

3/10/0614 Plane Wave Method Inserting Fourier series expansions in differential equation, assuming harmonic time dependence For M terms kept in the sum, this is an MxM matrix. Eigenvalues are n=1 (first band), 2 (second band) … Scanning Brillouin zone yields -Dispersion Relation -Equifrequency Surfaces (EFS) BZ

3/10/0615 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

3/10/0616 PW Prediction for Steel-Air Strong Band Gaps at kHz kHz What happens here? f=0.55 f=0.3

3/10/0617 PW Method For f >.8 Eigenvalues are imaginary Solutions would be of the form Introducing a damping/diverging term Real Physics? Probably not

3/10/0618 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)

3/10/0619 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 -source -scattered waves from all other scatterers scatterers Allows amplitude of field to be calculated at any point

3/10/0620 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 Bessel functions J n are used to ensure p does not diverge at center of cylinder

3/10/0621 Multiple Scattering Method Coefficients A i n and B i n 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 And a structure constant that depends on geometry of scatterers, Source term (2n+1)N x (2n+1)N matrix equation

3/10/0622 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 TheoryExperiment -Without crystal -With crystal Figures: Chen et. al 2003, Robertson 1998

3/10/0623 Difference with Nmax Nmax=1 Nmax=2 Nmax=4 Nmax=3 1GB Memory not enough for Nmax = 5

3/10/0624 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)

3/10/0625 Negative Refraction in Sonic Crystals PR/NR--Positive/Negative Refraction SC/PC--Sonic/Photonic Crystal 2 Types of Negative Refraction 1) Backward 2) Forward Figure: Feng, Acoustic Backward-Wave Negative Refraction 2006

3/10/0626 Backward Negative Refraction Steel Rods immersed in Water Figure: Sukhovich, Negative Refraction of ultrasonic waves

3/10/0627 Forward Negative Refraction Forward Negative Refraction can occur with Brillouin zones of model Photonic Crystal In regions of negative phononic effective mass Regions around M point have (Green Triangle)

3/10/0628 Simulating PWNR X. 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 (below this line) Figures: X. Zhang et al. (2004) Results are shown for mercury/water crystals For Steel Air systems? Mercury (EFS)

3/10/ 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)

3/10/0630 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 ~ EFS are roughly circular Can define effective refractive index (ERI) Figures: Zhang. et al. (2005) Use ERI in Snell’s Law, Brewster’s angle etc. f=0.47

3/10/0631 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

3/10/0632 Using negative ERI for imaging For the EFS at and The ERI is Brewster’s Angle is given by Near axis approximation gives Thickness of slab - # layers - radius of cylinder Expect rays to converge near I’ and I but “out of focus”

3/10/0633 Simulation Results n=-.7 PredictedSimulation result For Distance 9 layer sample-- d=7.76a 10.2a 18.2a 10.1a* 19.3a Predicted Simulation result For Distance 15 layer sample-- d=12.76a 10.2a 31.5a 10.1a* 33.8a* Predicted For Distance 15 layer sample-- d=12.76a 18.7a 31.5a 18.5a 33.8a Simulation result * * Independent of * Intensity

3/10/0634 Simulations for n=-1 For f=0.906 EFS at and The ERI is Layers For all images d=5.33a D 3 =10.8a d=6.20a D 3 =12.5a d=7.93a D 3 =16.1a Plots are of pressure: Source in phase with image D 1 fixed----Vary Thickness

3/10/0635 Simulations for n=-1 For f=0.906 EFS at and The ERI is 6 layers for all images D 1 =0.5a D 3 =10.8a Plots are of pressure: Source in phase with image D 1 =2.5a D 1 =4.5a D 3 independent of D 1 Thickness fixed----Vary D 1

3/10/0636 Reproducing results For n=-.7 pattern is similar, shows focusing effect

3/10/0637 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

3/10/0638 Additional Slides