Research ArticlePHYSICS

Topological transport of sound mediated by spin-redirection geometric phase

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Science Advances  16 Feb 2018:
Vol. 4, no. 2, eaaq1475
DOI: 10.1126/sciadv.aaq1475
  • Fig. 1 Sound vortices inside a circular waveguide.

    The solid lines show the dispersion relations of the lowest-order guided modes (monopole, dipole, and quadrupole), labeled by the corresponding mode pressure field. A vortex of charge q = +1 can be excited using four monopole sources with initial phases 0, π/2, π, and 3π/2, and the corresponding pressure field is shown in the inset. The vortex with q = −1 can be excited using monopole sources with a reversed order of arrangement.

  • Fig. 2 Spin-redirection geometric phase induced by helical transport of sound vortices.

    (A) Sound vortices traveling through a helical waveguide acquire a geometric phase in addition to the normal dynamic phase. Such a phase can be considered as the result of the rotation of the local coordinate system (u, v, t) attached to the waveguide. (B) Band structures for a straight waveguide and a helical waveguide showing the degeneracy lifting of the q = ±1 vortices due to the coupling of the OAM and linear momentum. The monopole band is not shown. (C) The degeneracy lifting can be attributed to the spin-redirection geometric phase induced by the evolution of the vortex states in wave vector space. The violet dot at the origin denotes the effective magnetic monopole that depends on the topological charge of the acoustic vortex. The circulation direction of k depends on the handedness of the helical waveguide. The red circle denotes the circulating direction of the OAM carried by the sound vortex.

  • Fig. 3 Experimental demonstration of the spin-redirection geometric phase for a sound vortex.

    (A to C) Experimental setups for a straight waveguide, a bent waveguide, and a helical waveguide, respectively. (D to F) Magnitude and phase of the pressure field detected at the end of the waveguide corresponding to (A) to (C), respectively. The sound vortex is excited using a monopole source array at the input. The dashed circles denote the boundary of the waveguide cross section. a.u., arbitrary unit; deg, degree.

  • Fig. 4 Rotation effect induced by the spin-redirection geometric phase.

    Comparison between the experimental and full-wave simulation results for the normalized magnitude of the input and output pressure fields when a linear dipole mode along the y axis is excited. The system configuration is the same as in Fig. 3C.

  • Fig. 5 Acoustic interferometer based on spin-redirection geometric phases.

    (A) Three-dimensional acoustic interferometer consisting of a helical waveguide and a bent waveguide. (B) Transmission contrast η (see main text for definition) of the q = ±1 vortices as a function of frequency for the 2D (not shown) and 3D interferometers. The 2D setup is obtained by unwinding the helical waveguide so that the two waveguides lie on a 2D plane. (C) Normalized magnitude and phase of the output pressure field of the q = ±1 vortices at the frequency f = 3295 Hz for the 3D case. (D) Normalized magnitude and phase of the output pressure field of the q = ±1 vortices at the frequency f = 3295 Hz for the 2D case, which has mirror symmetry.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/4/2/eaaq1475/DC1

    Mapping between waveguide dispersion and band structure

    fig. S1. Band structures for helical waveguides with different pitches.

    fig. S2. An acoustic interferometer based on spin-redirection geometric phases.

    fig. S3. Full-wave simulations of acoustic interferometers based on spin-redirection geometric phases.

  • Supplementary Materials

    This PDF file includes:

    • Mapping between waveguide dispersion and band structure
    • fig. S1. Band structures for helical waveguides with different pitches.
    • fig. S2. An acoustic interferometer based on spin-redirection geometric phases.
    • fig. S3. Full-wave simulations of acoustic interferometers based on spin-redirection geometric phases.

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