Negative refraction of ultrasonic waves in 2D phononic crystals.

Transcription

1 Negative refraction of ultrasonic waves in 2D phononic crystals. Alexey Sukhovich 1, John Page 1, Zhengyou Liu 2, Maria Kafesaki 3 1 University of Manitoba, Canada; 2 Wuhan University, China; 3 IESL-FORTH, Heraklion, Crete, Greece For more information on our research, go to

2 Left-Handed Materials (LHMs) and Negative Refraction Right-Handed materials: k and S are in same direction H S E k - normal refraction n = εµ > 0 Left-Handed materials: k and S are in opposite directions ε, µ > 0 [ k H ] = - ωε E ε, µ < 0 [ k E ] = ωµ H S = [ E H ] Veselago 1 predicted the unusual phenomenon of negative refraction in LHMs : k H S E - negative refraction n = εµ < 0 θ i ε 1, µ 1 > 0 θ i ε 1, µ 1 > 0 ε 2, µ 2 > 0 ε 2, µ 2 < 0 1 V. Veselago, Sov.Phys.Usp. 10, 509 (1968) θ r θ r Can we observe negative refraction of sound waves?

3 2D Phononic Crystals Our 2D phononic crystals: stainless steel rods of 1.02 mm in diameter assembled in 2D triangular crystal lattice with φ = 0.58 and immersed in water Rectangular-shaped 6-layer crystal : Prism-shaped : <10> <11> 60 <10> a 1 <11> a 2 30

4 Basic Properties : Transmission Coefficient TRANSMISSION EPERIMENT Rectangular Crystal: - measure the transmission coefficient of ultrasound waves through rectangular crystal incident along <11> direction - theoretical calculations by FDTD theory [M. Kafesaki] predict existence of a stop band along ΓΧ direction around 0.6 MHz - reasonably good agreement between theory and experiment TRANSMITTED INTENSIT Intensity transmission coefficient Experimental results: 6-LAYER STEEL ROD CRYSTAL 1E FREQUENCY (MHz) Intensity transmission coefficient Comparison with FDTD theory:

5 TRANSMISSION EPERIMENTS Rectangular Crystal: - by measuring phase difference between reference Γ and transmitted pulses, the dispersion relation can be found experimentally determined band-structure of the crystal - Multiple Scattering Theory (MST) [Liu et al. PRB 62, 2446 (2000)] theoretically predicted band-structure Theory and Experiment : GOOD AGREEMENT 10 OMEGA " (rads/µs) Basic Properties :Band Structure BAND STRUCTURE: MST CALCULATIONS! #! WAVE VECTOR k (1/mm) FREQUENCY (MHz) 10 OMEGA # (rads/µs) " Χ Μ EPERIMENT MST CALCULATIONS WAVE VECTOR k (1/mm)!

6 Band Structure and Negative Refraction in Phononic Crystals Region of interest 2 nd band FREQUENCY f (MHz) ! "! WAVE VECTOR k (1/mm) r v g k red v g - for waves along ΓΧ direction group and phase velocities are in opposite directions since r = "!(k) k MST also allows equifrequency contours be theoretically calculated k y a nd BAND MST EQUIFREQUENCY CONTOURS: 0.95 MHz 0.85 MHz 0.75 MHz f = 0.75 MHz f = 0.85 MHz f = 0.95 MHz k red!g " M - band structure leads to an effect which looks like negative refraction in LHMs k x a

7 Experimental Set-up - narrow Gaussian ultrasound pulse incident normally at the prism-shaped crystal along ΓΧ direction - hydrophone ( miniature ultrasound transducer element diameter < λ ) scans transmitted field in a plane perpendicular to the rods ( Y plane ) 2D image plot of wave field at the output side - digitally filter measured field with a narrow Gaussian bandwidth to obtain the wave field at different frequencies Y OUTPUT SIDE GENERATING TRANSDUCER INCIDENT WAVE 60 M INPUT SIDE Γ MOTORIZED STAGE Z HYDROPHONE 30

8 Experimental Set-up QUESTION : Will we see positive or negative refraction? GENERATING TRANSDUCER INCIDENT WAVE Negative 60 M INPUT SIDE Γ Positive 30

9 Experimental Results 1 Middle of the 2 nd band: f = 0.85 MHz NEGATIVE REFRACTION! As predicted when v g and k are antiparallel a.u.

10 Experimental Results 2 Lower frequency : f = 0.75 MHz NEGATIVE REFRACTION AND POSITIVE REFRACTION a.u.

11 Interpretation : INPUT side - normally incident wave enters crystal without change in its original direction - excites a Bloch wave inside the crystal with two dominant wave vectors: ETENDED zone scheme: k ext = k red + b 1 - parallel to v g (b 1 reciprocal wavevector) REDUCED zone scheme: k red - antiparallel to v g 7 RADIAL FREQUENCY # (mm/µs) " k red 1 st Brillouin zone! b 1 /2 k ext 2 nd Brillouin zone WAVE VECTOR FREQUENCY (MHz) v g k ext k wat k red v g Γ b 1 Both waves propagate in the same direction! M

12 Interpretation : OUTPUT Side - component of the incident wave vector k parallel to the interface is conserved ( Snell s law) - expect TWO types of refraction simultaneously: positive and negative M k wat Γ M k wat Γ k ext 60 k wat α k red 60 α k wat Snell s law: k crystal sin(60 ) = k wat sin(α)

13 Experimental Results f = 0.75 MHz f = 0.85 MHz α β α Theory and Experiment: Excellent Agreement k red = 2.09 mm -1 k ext = 3.62 mm -1 Negative refraction angle α : Predict: 35 Observed: 34 ± 1 Positive refraction angle β : Predict: 81.9 Observed: 81 ± 1 k red = 1.44 mm -1 k ext = 4.27 mm -1 Negative refraction angle α : Predict: 20.4 Observed: 21 ± 1 No Positively refracted beam observed Snell s law predicts total internal reflection

14 Reversed Experiment Angle of incidence 20 w.r.t. ΓΧ direction- same as the angle at which 0.85 MHz wave emerged in the direct experiment EPECT: 0.85 MHz wave should emerge normally in the reverse experiment Predict refraction angle for other frequencies using : Y Equifrequency Contours b 1 k ext v g Γ k red v g k wat M 60 α Γ 20 M Snell s law

15 Reversed Experiment : Results f = 0.85 MHz f = 0.75 MHz Wave emerges normally as expected α Expected : 18.6 Observed : 19 ± 1

16 Conclusions We have experimentally demonstrated NEGATIVE REFRACTION of ultrasound waves in a 2D phononic crystal. Because of the phononic crystal band structure, v g and k are antiparallel in the 2 nd band (reduced zone scheme). waves refract negatively. Both Negatively and Positively refracted waves are observed at some frequencies (the Bloch wave inside the crystal has two dominant wavevectors). MST and Snell s law allows the angles of refraction to be calculated theoretically Excellent agreement with the experiment! Circular equifrequency contours in the 2 nd band allow the possibility of observing focusing of sound by a flat 2D phononic crystal acoustic lens.

17 Focusing of Ultrasound by a Flat Phononic Crystal point source θ i v g v g θ r phononic crystal k ext k wat v g Γ M k red

18 Focusing of Ultrasound by a Flat Phononic Crystal Focusing at 0.75 MHz source Pinducer source (2.4-mm-diameter) Circular equifrequency contours in the 2 nd band FWHM of focal spot 1.5 λ Effective refractive index : n r, eff k crystal i = = = " k water FOCUS WIDTH sin! sin! r 0.66 λ