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(a) Band structure of PC generated by PWE method (Ref. 6) (b) Extended zone scheme and EFCs of air evaluated at 13.5 kHz (in units of ). An incident acoustic signal with angle impinging upon the crystal excites multiple Bloch modes via reciprocal lattice vector translation of a -vector located within the first Brillouin zone. In the PC, the gray arrows represent wave vectors and the black arrows represent group velocity vectors. Bloch waves with parallel -vector component coinciding with the EFC for air on the exit contribute to beam splitting. (c) Schematic representation of FDTD simulation domain.
Phase shift per PC length in reference to the zero-angle refraction angle of 28.1°. The solid black dots represent analytical results from Eq. (1) and the unfilled triangles represent FDTD calculations. The FDTD results are obtained with a crystal of length 540 mm (20 lattice parameters). The uncertainty on calculated phase difference in FDTD calculations is 0.027 radians.
(a) FDTD instantaneous pressure plot of the simulation domain with two incident beams at 10° and 28.1° impinging on a PC with dimensions ( lattice parameters). The time average of the absolute value of instantaneous pressure over one period (referred to as average pressure) is calculated along two spatial cuts indicated as solid black lines. (b) Profile of average pressure taken along cuts in (a). (c) FDTD instantaneous pressure plot of simulation domain with two incident beams at 10° and 28.1° impinging on PC with dimensions ( lattice parameters). (d) Profile of average pressure taken along cuts in (c).
FDTD instantaneous pressure plot of simulation domain with two incident beams at −32° and 24° impinging on PC with dimensions ( lattice parameters). The input waves excite the same Bloch waves within the PC. For a radian phase shift between the incident beams there is destructive interference along all paths of travel (a) and for zero phase shift there is constructive interference along all paths of travel (b).
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