Fig. 2. Curved parallel-plate waveguide.

a Electric field solution (eigenvector) for the fundamental transverse electric (TE₁)-like mode for 200 GHz (top) and 280 GHz (bottom), for radii of 0.3 mm (blue) and 3 mm (red) compared to a planar parallel-plate waveguide (black dashed). The insets show the electric field propagation obtained with a finite element method simulation with a radius of 0.5 mm for 200 GHz (top) and 280 GHz (bottom) (in red and blue corresponding to positive and negative values of the electric field). b Effective refractive index n_eff as a function of the radius of curvature for various frequencies. The dots are the calculated values obtained from the eigensolution, while the lines correspond to the fits (Eq. 2). The dashed lines correspond to the effective refractive index for a planar parallel-plate waveguide. The gray line is 10R_c and distinguishes the regimes of curved and planar behavior (as explained in the text). c Comparison of the critical radii obtained with the fit to the expression obtained from geometrical optics. The critical radius obtained from the fit can be fitted to a fourth-order polynomial: $10R_c^{\mathrm{fit}}\left[\mathrm{mm}\right]=-9368f^4+9993f^3-3983f^2+715f-45$, with the frequency f expressed in units of [THz]. The geometrical optics view of propagation in a parallel-plate waveguide and a curved waveguide are shown in the top and bottom insets, respectively. In the bottom inset, both regimes of alternating reflections (red ray) and whispering-gallery mode reflections (blue ray) are depicted. The black ray corresponds to the critical angle (as explained in the text). d Experimentally measured effective refractive index (dots) in the single-mode operation range (150–300 GHz, b = 1 mm) compared to the analytical models (bold lines) for a conformal waveguide with R = 3.5 mm (blue) and a planar waveguide (red).