Fig. 1: Illustration of the phase matching in SPPS and schematics of two potential realizations.

a SPPS interaction in the optical (straight solid) and polaritonic (curved solid) dispersion relation: the pump (green point, frequency ω, and wave number k) is scattered into the counter-propagating Stokes mode (red point) where the difference in frequency and momentum (blue line) matches the polaritonic dispersion relation. This polariton mode (blue point, wave number q ≈ 2k for angular frequency Ω ≪ ω) is amplified along the waveguide. b Conceptual schematic of an SPPS experiment in Stokes-seed configuration. A weak Stokes seed (red arrows) is injected at the rear end, and amplified as it propagates toward the front end of a graphene-covered waveguide. This action is also illustrated by qualitative spectra where the heights of the colored peaks indicate to the relative amplitudes of the matching signals at the front and back of the waveguide. The bias voltage contact and the waveguide geometry shown here were intentionally kept oversimplified at this stage and in reality would require careful engineering. This configuration is best suited for the excitation of polaritons and for characterizing their dispersion relation, which can be tuned via the bias voltage contact. It is also the natural candidate for a first demonstration of SPPS, as all input and output signals are in the optical domain and no injection or detection of THz waves is required. c Conceptual schematic of an SPPS experiment in polariton-seed configuration. A weak THz signal is injected through the bias contact (blue), and amplified along the waveguide. Simultaneously, an optical Stokes is generated (illustrated by qualitative spectra similar to panel b). This configuration is compelling because of the prospect to amplify or optically detect weak THz signals.