  Planar waveguides can limit the transmission of electromagnetic waves in a
specific direction and have a wide range of applications in filters, sensors,
and energy-transfer devices. However, given the increasing demand for planar
integrated photonics, new waveguides are required with excellent
characteristics such as more functionality, greater efficiency, and smaller
size. In this work, we report the experimental results for a subwavelength
planar microwave-regime waveguide fabricated from circuit-based magnetic
hyperbolic metamaterial (HMM) and develop the associated theory. HMM is a
special type of anisotropic metamaterial whose isofrequency contour (IFC) takes
the form of an open hyperboloid. Because of the open-dispersion IFCs, HMMs
support propagating high-k modes with large effective refractive indices, which
allows planar hyperbolic waveguides to be miniaturized. In particular, as
opposed to the traditional dielectric slab waveguide, the group velocity of the
hyperbolic guided modes is negative-a characteristic that can be exploited to
realize the backward propagation of electromagnetic waves. Furthermore, we
study the abnormal photonic spin Hall effect and experimentally demonstrate an
ultracompact backward coupler based on hyperbolic waveguides. The experimental
results are consistent with numerical simulations. This work not only reveals
the significant potential of circuit-based platforms for the experimental study
of the propagation and coupling of guided modes but also promotes the use of
HMMs in the microwave regime for numerous integrated functional devices.
