Fig. 1. Schematic illustration of third-harmonic generation and second-harmonic generation processes in inverse-designed microstructured fibers and metasurfaces, respectively.

Fig. 2. Dispersion relations (solid line) and radiative lifetimes Q (dashed line) versus propagation wavenumber k of TM01 fundamental ω1 (red) and third-harmonic ω3 (blue) modes in a chalcogenide/PES fiber optimized to achieve frequency matching ω3=3ω1 and large nonlinear overlaps at kopt=1.4(2π/λ). The shaded area in gray indicates regions lying below the chalcogenide light cone. The top insets show the fiber cross section overlaid with corresponding power densities at ω1 (left) and ω3 (right).

Fig. 3. Nonlinear overlap factor |β3|2 corresponding to fundamental and third-harmonic modes in fibers that have been optimized to ensure phase-matched modes (k3=3kopt) at various fundamental-mode propagation wavenumbers kopt, for both TE01 (blue) and TM01 (red) polarizations, by the application of either topology (circles or squares) or shape (triangles) optimization. The gray-shaded area denotes the regime of guided modes below the chalcogenide lightline. For comparison, also shown is |β3|2 (black cross) of a standard plain fiber manually designed for operation at ω1=0.914(2πc/λ) and k1=0.992(2π/λ) [77]. Shown as insets are fiber cross sections along with power densities of fundamental modes optimized at four different kopt={0.1,1.4,1.7,2.0}(2π/λ) for both TE01 (upper insets) and TM01 (lower insets), with (i)–(iii) obtained via topology optimization and (iv) via shape optimization.

Fig. 4. (a) Schematic illustration of second-harmonic generation in a square-lattice metasurface of finite thickness t and period Λ×Λ. Shown to the right are dielectric profiles and mode profiles |E|2 corresponding to two inverse-designed metasurfaces, both over single unit cells and z=0 cross sections. The structures are optimized to ensure frequency and phase matching for light incident at (i) an angle θ=3° or (ii) normal incidence. Dark (white) represents gallium phosphide (vacuum) regions. (b) Convergence of the objective function with respect to iteration number, leading to structure (ii).

Table 1. Representative Second-Harmonic Generation FOMs for Both Hand- and Inverse-Designed Metasurfaces, Including χ(2), Fundamental Wavelength λ1, and Conversion Efficiency η per Unit Cell^a