  The suite of highly confined polaritons supported by two-dimensional (2D)
materials constitutes a versatile platform for nano-optics, offering the means
to channel light on deep-subwavelength scales. Graphene, in particular, has
attracted considerable interest due to its ability to support long-lived
plasmons that can be actively tuned via electrical gating. While the excellent
optoelectronic properties of graphene are widely exploited in plasmonics, its
mechanical flexibility remains relatively underexplored in the same context.
Here, we present a semi-analytical formalism to describe plasmons and other
polaritons supported in waveguides formed by bending a 2D material into a
parabolic shape. Specifically, for graphene parabolas, our theory reveals that
the already large field confinement associated with graphene plasmons can be
substantially increased by bending an otherwise flat graphene sheet into a
parabola shape, thereby forming a plasmonic waveguide without introducing
potentially lossy edge terminations via patterning. Further, we show that the
high field confinement associated with such channel polaritons in 2D parabolic
waveguides can enhance the spontaneous emission rate of a quantum emitter near
the parabola vertex. Our findings apply generally to 2D polaritons in
atomically thin materials deposited onto grooves or wedges prepared on a
substrate or freely suspended in a quasi-parabolic (catenary) shape. We
envision that both the optoelectronic and mechanical flexibility of 2D
materials can be harnessed in tandem to produce 2D channel polaritons with
versatile properties that can be applied to a wide range of nano-optics
functionalities, including subwavelength polaritonic circuitry and bright
single-photon sources.
