Thermally Stable Si₃N₄ Photonic Circuits Enabled by TiO₂ Overlays and dn/dT Measurements

We present a simulation framework to engineer athermal integrated waveguides in Si₃N₄ photonic platforms using Si₃N₄/TiO2/SiO₂ material stacks. Starting from realistic cross-sections, we compute the temperature dependence of guided modes by combining electromagnetic eigenmode solvers with temperature-dependent refractive-index models. The thermo-optic dispersion is constrained using our new measurements of dn/dT for TiO₂ and Si₃N₄, enabling more accurate prediction of the effective-index shift. Thermo-optic finite-element simulations provide the mode distribution, allowing us to separate and quantify contributions to Δneff(T) from the Si₃N₄/TiO₂/SiO₂ core, TiO₂ layer, and SiO₂ cladding. By sweeping core geometry and TiO₂ thickness within the SiO₂ clad environment, we map the design space where the net thermo-optic coefficient of the fundamental mode approaches or crosses zero, and we quantify the spectral bandwidth over which this athermal condition is maintained. The methodology provides a practical, simulation-driven route to identify and optimize athermal Si₃N₄/TiO₂/SiO₂ waveguide geometries for temperature-stable operation in precision photonic integrated circuits.