Fig. 1: Design and simulation of the χ⁽²⁾ waveguide with birefringence PM based on a ZGP crystal.

a Calculated normalized gain spectrum at a pump wavelength of 2.4 μm for a bulk ZnGeP₂ (ZGP) crystal, at various phase match (PM) angles. A PM angle of θ = 48.3° is designed as marked by the orange dashed line to generate the broadest gain bandwidth. b Simulated electric field distributions in the χ⁽²⁾ waveguide of pump wave at 2.4 μm in TM₀₀ mode (top) and a typical idler wave at 8.0 μm in TE₀₀ mode (bottom). c Calculated propagation losses of fundamental modes in the designed χ⁽²⁾ waveguide in TE and TM polarizations with the angle between the propagation direction and the crystal optical axis as 48.3°, in a broad spectral range of 2–11 μm. The results indicate that the propagation loss of fundamental TE and TM mode is less than 1.5 dB/cm and 4.0 dB/cm, respectively. Notably, calculated propagation losses of fundamental modes are <0.05 dB/cm at 8 μm, and <0.25 dB/cm (TE) and 0.5 dB/cm (TM) at 9 μm, respectively. The raise of waveguide losses in the wavelength range of 9–11 μm is attributed to the absorption of fused silica substrate peaked at 9.5 μm, as shown in the top inset which depicts the absorption coefficient (imaginary part of complex refractive index) of silica, as a function of wavelength²². Meanwhile, the material loss of ZGP crystal is minimal in the spectral range of 2–11 μm as shown by the measured transmission spectrum of a 10-mm-thick uncoated ZGP crystal in the bottom inset of c. d The simulated group velocity dispersion (GVD) of TM₀₀ modes in a wavelength range of 2 to 11 μm, in the designed χ⁽²⁾ ZGP waveguide (green) and bulk ZGP (black), respectively. Owing to multi-wavelength-scale dimensions, the nonlinear waveguide exhibits nearly an identical GVD profile to that of the bulk material, which simplifies the design of χ⁽²⁾ waveguide devices with birefringence PM.