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

This paper introduces a full-wave harmonic-balance framework that accurately models dispersive time-varying photonic structures by representing temporal modulation as off-diagonal coupling in Floquet space, enabling the design and optimization of devices like graphene cylinders and ENZ metasurfaces for applications such as frequency conversion and nonreciprocity.

Original authors: Mohammad R. Tavakol, Wenshan Cai

Published 2026-08-11

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CC BY 4.0

Original authors: Mohammad R. Tavakol, Wenshan Cai

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the world of light not as a steady beam, but as a bustling dance floor where energy is constantly being swapped. In the old days, scientists thought of light and matter as static partners: a mirror reflects light, a lens bends it, and the color (frequency) stays exactly the same. But recently, researchers discovered that if you wiggle the material itself—changing its properties faster than the light can blink—you can break the rules. You can make light change color, bounce in weird directions, or even get amplified without magnets. This is the exciting, chaotic world of "time-varying photonics."

However, there's a catch. Real-world materials, like special metals or graphene, are "disperse," meaning they react differently to different colors of light, almost like a sponge that soaks up water at different rates depending on the temperature. When you try to wiggle these tricky materials, the math gets incredibly messy. Every new color of light created by the wiggling interacts with the material in a unique way, and they all talk to each other at the same time. Trying to simulate this is like trying to predict the movement of a thousand dancers who are all holding hands, changing partners, and reacting to a drumbeat that keeps speeding up. Until now, scientists have struggled to build a computer model that can handle this complexity without getting lost in the noise.

This paper introduces a clever new tool called a "Full-Wave Harmonic Balance Framework" to solve that exact problem. Think of it as a super-smart translator that takes a chaotic, time-wiggling light show and breaks it down into a neat, organized grid of steady-state snapshots. Instead of watching the light dance frame-by-frame (which takes forever and is prone to errors), this method looks at all the "harmonics"—the original light plus all the new colors it creates—simultaneously. It treats the wiggling material not as a moving target, but as a generator of invisible "secondary sources" (like tiny, hidden batteries) that push and pull the light into these new colors.

The authors tested this new framework on two very different scenarios to prove it works. First, they simulated a cylinder wrapped in graphene (a super-thin carbon material) where the electrical properties were wiggled at a specific speed. They found that their method perfectly predicted how the light would scatter and get absorbed, matching up with known math formulas. Second, they designed a high-tech mirror made of a special glass-like material (ITO) that changes its properties when hit by a pump. They used their framework to optimize the shape of this mirror so that it could take an incoming beam of light, change its color, and bounce it off at a precise new angle. The results showed that this new approach can accurately model how energy is shuffled between different colors and directions in complex, real-world materials, offering a powerful new way to design future devices that control light with the speed and precision of a conductor leading an orchestra.

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

Problem Statement
Time-varying photonic structures offer unique capabilities such as frequency conversion, nonreciprocity, and dynamic wavefront control by breaking temporal translation symmetry. However, accurately modeling these systems remains a significant challenge when the materials involved are strongly dispersive (e.g., transparent conducting oxides, graphene, epsilon-near-zero materials). In such platforms, temporal modulation cannot be treated as a simple instantaneous perturbation of a frequency-independent parameter. Instead, causality dictates that the material response depends on the excitation history, meaning each generated Floquet harmonic experiences a distinct material response while remaining coupled to all other harmonics through the modulation. Existing computational methods face trade-offs: time-domain methods (like FDTD) often require long simulations to reach steady-state and struggle with causal, dispersive updates, while specialized analytical or coupled-wave methods often lack the geometric flexibility to handle complex 3D structures, inhomogeneous media, or arbitrary modulation spectra.

Methodology
The authors propose a full-wave harmonic-balance framework that solves for the steady-state electromagnetic response directly in the frequency domain, avoiding time-marching. The core of the methodology involves recasting the time-varying problem as a coupled set of frequency-domain subproblems within an extended Floquet space.

Key elements of the formulation include:

Secondary Source Representation: Time-varying components are represented not as time-dependent boundary conditions, but as induced secondary sources. Volumetric polarization densities ( $P$ ) are used for modulated bulk media, and surface current densities ( $J_s$ ) are used for modulated conductive sheets.
Floquet Expansion: All fields and induced sources are expanded into a truncated Floquet series ( $F(r, t) = \sum F_n(r)e^{j\omega_n t}$ ), where $\omega_n = \omega_0 + n\Omega$ .
Coupled System Formulation: The problem is solved as a global linear system $(L - SC)X = Y_{ext}$ $(L - S C) X = Y_{e x t}$ .
- $L$ represents the block frequency-domain Maxwell operator for the static background, evaluated at the physical frequency of each harmonic ( $\omega_n$ ) to account for material dispersion.
- $S$ represents source-injection operators.
- $C$ represents the source-generation operators derived from the temporal Fourier expansion of the material parameters. These operators introduce off-diagonal coupling in the harmonic space, representing the frequency conversion mechanism.
Implementation: The framework is implemented using the Finite Element Method (FEM). The standard FEM unknown vector is lifted into a harmonic-domain vector, allowing each spatial degree of freedom to carry multiple harmonic coefficients. This preserves the geometric flexibility of full-wave modeling while capturing the inter-harmonic coupling.

Key Contributions

Unified Framework: The method provides a single algebraic structure to treat both modulated volumetric media and modulated conductive sheets, supporting arbitrary temporal modulation spectra (single-tone, multi-tone, or non-sinusoidal).
Dispersion Handling: Unlike methods that assume weak dispersion or use envelope approximations, this framework evaluates material dispersion and radiation operators at the specific frequency of each generated harmonic.
Harmonic-Resolved Observables: The solution directly yields steady-state amplitudes, phases, and field profiles for all retained Floquet harmonics, enabling the calculation of harmonic-specific scattering, absorption, power flow, and near-field distributions without post-processing time-domain data.

Results
The framework was validated and demonstrated through two representative examples:

Parametrically Pumped Rolled Graphene Cylinder:
- Setup: A dielectric cylinder wrapped in a graphene sheet with a temporally modulated Fermi level ( $\Omega = 2\omega_0$ ), operating in a degenerate parametric amplification regime.
- Validation: The numerical results for harmonic-resolved scattering and absorption showed excellent agreement with an analytical Floquet scattering solution derived for the cylindrical geometry.
- Findings: The simulation successfully captured the redistribution of energy from the carrier to the down-converted harmonic ( $n=-1$ ), confirming the framework's ability to model strong dispersion and rapid modulation in resonant surface plasmon systems.
Optical Space-Time Metasurface (ITO-based):
- Setup: A reflective metasurface consisting of a binary $TiO_2$ grating on a time-modulated Indium Tin Oxide (ITO) epsilon-near-zero (ENZ) layer backed by gold. The ITO plasma frequency was modulated to drive ultrafast changes in permittivity.
- Application: The framework was used to design and optimize the metasurface for frequency-converting beam deflection. The goal was to convert an incident optical wave into sidebands and redirect them into a specific spatial diffraction order ( $m=1$ ).
- Findings: The optimized design successfully redirected the generated sidebands ( $\omega_{\pm 1}$ ) into the first spatial diffraction order. The results demonstrated the interplay between ENZ dispersion, time-dependent Drude coupling, and spatial diffraction, validating the method's utility for designing complex, realistic optical devices.

Significance
The paper claims that this framework establishes a general platform for modeling and designing dispersive active photonic structures. By combining the physical transparency of harmonic-balance approaches with the geometric flexibility of full-wave FEM, the method overcomes the limitations of existing solvers for time-varying systems. It allows researchers to directly quantify how modulation redistributes electromagnetic energy among frequency-shifted channels, providing essential observables (harmonic-resolved power, absorption, and diffraction) for the design of next-generation space-time photonic systems based on graphene, transparent conducting oxides, and resonant metasurfaces. The work positions the framework as a tool not only for analysis but also for the inverse design of devices requiring frequency conversion, parametric amplification, and dynamic wavefront control.

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