  We report direct visualization of gigahertz-frequency Lamb waves propagation
in aluminum nitride phononic circuits by transmission-mode microwave impedance
microscopy (TMIM). Consistent with the finite-element modeling, the acoustic
eigenmodes in both a horn-shaped coupler and a sub-wavelength waveguide are
revealed in the TMIM images. Using fast Fourier transform filtering, we
quantitatively analyze the acoustic loss of individual Lamb modes along the
waveguide and the power coupling coefficient between the waveguide and the
parabolic couplers. Our work provides insightful information on the
propagation, mode conversion, and attenuation of acoustic waves in
piezoelectric nanostructures, which is highly desirable for designing and
optimizing phononic devices for microwave signal processing and quantum
information transduction.
