  In this work, a parameterized eigenvalue problem is analyzed for a phononic
array in a 2D stress wave scattering setup, and a corresponding sensing
application of this system is proposed to achieve source angle localization.
The phononic domain consists of a periodic micro-structured medium, of which
the eigen-wavevector band structure and the eigen-modes are exploited. The
eigen-modes are naturally angle dependent due to changes in phases and periodic
mode shapes determined by the incident angle. Intriguingly, the band exhibits
angle-dependent transitions at the exceptional points (EPs) and critical angles
(CAs), where the eigenvalues coincide or vanish. Coupled with these
transitions, it is found that the eigen-modes switch their energy
characteristics and symmetry patterns at these branch points, leading to
enhanced angle dependence. Moreover, these eigen-modes also serve as the basis
functions of the scattered waves. Therefore, the scattering response of the
medium inherently possesses the angle-dependent properties, making this system
naturally suitable for sensing applications. An artificial neural network (ANN)
is trained with randomly weighted eigen-modes to achieve deep learning of the
eigen features and angle dependence. The training data is derived only based on
the eigen-modes of the unit cells. Nevertheless, the trained ANN can accurately
identify the incident angle of an unknown scattering signal, with minimal side
lobe levels and suppressed main lobe width. The ANN shows superior performance
in comparison with standard delay-and-sum technique of estimating angle of
arrival. The proposed application of ANN and micro-structured media highlights
the physical importance of band structure topology and eigen-modes to a
technological application, adds extra strength to the existing localization
methods, and can be easily enhanced with the fast-growing data-driven
techniques.
