(a) Phononic crystal consisting of stepped cylinders arranged in square lattice and coated on a thin plate. Enlarged plot shows the unit cell and dimensions. (b) Irreducible first Brillouin zone ΓXMΓ of square lattice in wavevector space. (c) Top view and (d) side view of the manufactured phononic plate without defects. (e) Manufactured phononic plate with line defects to serve as straight and 90° bent waveguides.
Scheme of the experimental setup of laser ultrasonic measurement.
Acoustic band structures of the square lattice phononic plate with periodic stepped cylinders of different neck diameters d 2. (a) d 2 = 7.5 mm (uniform cylinder), (b) d 2 = 7.1 mm, and (c) d 2 = 6.7 mm, where the lattice constant, plate thickness, head diameter, head length, and neck length are fixed at a = 10.0 mm, e = 1.0 mm, d 1 = 7.5 mm, h 1 = 4.0 mm, and h 2 = 1.0 mm, respectively.
Calculated transmission spectra of different plate modes incident into the finite phononic plates of six unit cells along ΓX ((a)–(c)) and ΓM ((d)–(f)) directions, respectively. The solid curves and dashed curves represent the case of stepped cylinders of neck diameter d 2 = 6.7 mm and the case of uniform cylinders of diameter d 2 = d 1 = 7.5 mm, respectively. (a)–(c) Incident waves are S0, SH0, and A0 plate modes along the ΓX direction, respectively. (d)–(f) Incident waves are S0, SH0, and A0 plate modes along the ΓM direction, respectively.
Measured reference spectra for pulsed laser generated ultrasonic waves in the phononic plate propagating along the ΓX (a) and ΓM (b) directions, respectively. Solid and dashed curves represent the measurements using the longitudinal thetransverse transducers, respectively. The transducers are located on the top surface of a stepped cylinder to receive the wave signals. Five stepped cylinders lie in between the pulsed laser source spot and the transducer receivers. The acoustic band structures of corresponding propagation directions are shown above for comparison. The frequency ranges marked as gray denote the band gaps along the direction.
Calculated defect bands of line-defect phononic plate with stepped cylinders in the absolute band gaps (the unshaded frequency ranges). One row of cylinders is assumed to be removed from the perfect stepped cylinder array to form the line defect. (a) Defect bands in the first absolute band gap; (b) defect bands in the second and third absolute band gaps. The results are obtained using the supercell approximation method with nine unit cells contained.
Total displacement fields and mode shapes of several line-defect modes in the first and second absolute band gaps. The frequencies of modes A–H are correspondingly labeled in Fig. 6 . Modes A and B belong to the two defect bands in the first absolute band gap, where only mode A is well confined to the line-defect waveguide. Modes C–H belong to the defect bands in the second absolute band gap. All the modes are well confined by the line-defect waveguide, where mode F is an extensional plate mode, and the other modes are flexural modes.
Measured reference spectra of pulsed laser generated ultrasonic waves in the line-defect straight waveguide (Straight WG). The frequency ranges marked as gray denote the band gaps along the ΓX direction. Solid and dashed curves represent the measurements using the longitudinal the transverse transducers, respectively. The laser source spot and the transducer receivers are in the waveguide and separated by 7 a = 70 mm along the guiding path.
Calculated frequency responses of total displacement intensity with a sinusoidal point source applied in the middle of the waveguide. (a)Frequency f = 41.25 kHz (in the first absolute band gap), (b) f = 156.25 kHz (in the second absolute band gap), and (c) f = 103.75 kHz (outside the absolute band gaps). (d) The same as (c) with only the x-component amplitude of the displacement field.
Measured reference spectra of pulsed laser generated ultrasonic waves propagating in the 90° bent waveguide (Bent WG) at (a) point R1 and (b) point R2, respectively. The inset in (b) shows the locations of the laser source spot (S) and detection points (R1 and R2). The frequency ranges marked as gray denote the absolute band gaps. Solid and dashed curves represent the measurements using the longitudinal and transverse transducer receivers, respectively.
Calculated frequency responses of total displacement intensity with a sinusoidal point source applied in the front of the 90° bent waveguide. (a)–(c) Frequencies f = 54, 70, and 131 kHz (in between the first and second absolute band gaps), (d) f = 161 kHz (in the second absolute band gap), and (e) f = 39 kHz (in the first absolute band gap). (f) The same as (e) with two more stepped cylinder scatterers being removed from the bending corner.
Band-gap ranges and widths with different neck diameters. Geometry parameters: a = 10.0 mm, e = 1.0 mm, d 1 = 7.5 mm, h 1 = 4.0 mm, h 2 = 1.0 mm.
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