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.
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