(a) Scheme of U-shape waveguide design for frequency doubling. (b) Color map of the azimuthal number difference Δm between the TE₀₀ mode at ω₀ and the TM₀₀ mode at 2ω₀. (c) Δm vs the waveguide width for height h = 110 nm. Modal PM and $\overline{4}$-QPM conditions are reported in red and gray dashed line, respectively, while the markers correspond to PM. (d) Fraction of the power converted from FF to SH. The red line refers to the optimal situation where a change of the waveguide width enables to fulfill the $\overline{4}$-QPM condition in the curved section. The blue dashed line considers a constant width through the whole trajectory. (e) Spectral evolution of I_SHG. (f) Asymmetric SHG spectrum of a U-shaped χ⁽²⁾ waveguide.