Full-Wave Harmonic Balance Framework for Dispersive Time-Varying Photonic Structures

Time-varying photonic structures redistribute electromagnetic energy among Floquet harmonics, enabling frequency conversion, nonreciprocity, parametric gain, and dynamic wavefront control. Accurate modeling of realistic platforms remains challenging when temporal modulation occurs in strongly dispersive materials, because each harmonic experiences a distinct material response while remaining coupled to all others through the modulation. This work introduces a full-wave harmonic-balance framework for dispersive time-varying photonic structures. The formulation solves the steady-state Floquet response in the frequency domain by representing modulated components as induced secondary sources: volumetric polarization densities for bulk media and surface current densities for conductive sheets. Material dispersion and radiation operators are evaluated at each harmonic frequency, whereas temporal modulation enters as off-diagonal convolutional coupling in Floquet space. The framework is validated for a parametrically pumped rolled graphene cylinder, where numerical results agree with an analytical Floquet scattering solution and resolve harmonic-specific scattering, absorption, and near fields. It is further applied to an ITO-based epsilon-near-zero (ENZ) space-time metasurface, enabling shape optimization of a reflective device that converts an optical input into sidebands and redirects them into selected spatial diffraction orders. The approach establishes a general platform for modeling and designing dispersive active photonic structures with harmonic-resolved field, power, absorption, and diffraction observables.