Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Hamiltonian structure-preserving discretization of Maxwell's equations in nonlinear media

William Barham, Yaman Güçlü, Philip J. Morrison, Eric Sonnendrücker

TL;DR

This work develops a Hamiltonian framework for Maxwell's equations in nonlinear media and derives a structure-preserving finite element discretization via FEEC that exactly conserves Gauss's laws and yields a discrete energy bound. By specializing to isotropic cubic nonlinear media, the authors formulate a cubic Maxwell system with auxiliary dispersive variables, derive a field-independent Poisson bracket, and construct a discrete Hamiltonian that preserves the continuous invariants under a Strang-splitting time integrator. The spatial discretization leverages FEEC with dual de Rham complexes to ensure the discrete constitutive relations align with the energy structure, enabling exact Gauss law preservation and energy behavior that shadows the continuous system. Numerical results in 1D and 2D demonstrate optimal convergence, Casimir conservation, and accurate representation of nonlinear phenomena such as higher-harmonic generation and traveling-wave solutions, with explicit treatment of dissipative effects and time-dependent sources. The framework provides a robust, extensible approach for time-domain simulations of a broad class of nonlinear optical problems, with potential extensions to anisotropic media and broken-FEEC/HDG variants.

Abstract

A simple Hamiltonian modeling framework for general models in nonlinear optics is given. This framework is specialized to describe the Hamiltonian structure of electromagnetic phenomena in cubicly nonlinear optical media. The model has a simple Poisson bracket structure with the Hamiltonian encoding all of the nonlinear coupling of the fields. The field-independence of the Poisson bracket facilitates a straightforward Hamiltonian structure-preserving discretization using finite element exterior calculus. The generality and relative simplicity of this Hamiltonian framework makes it amenable for simulating a broad class of time-domain nonlinear optical problems. The main contribution of this work is a finite element discretization of Maxwell's equations in cubicly nonlinear media which is energy-stable and exactly conserves Gauss's laws. Moreover, this approach may be readily adapted to consider more general nonlinear media in subsequent work.

A Hamiltonian structure-preserving discretization of Maxwell's equations in nonlinear media

TL;DR

This work develops a Hamiltonian framework for Maxwell's equations in nonlinear media and derives a structure-preserving finite element discretization via FEEC that exactly conserves Gauss's laws and yields a discrete energy bound. By specializing to isotropic cubic nonlinear media, the authors formulate a cubic Maxwell system with auxiliary dispersive variables, derive a field-independent Poisson bracket, and construct a discrete Hamiltonian that preserves the continuous invariants under a Strang-splitting time integrator. The spatial discretization leverages FEEC with dual de Rham complexes to ensure the discrete constitutive relations align with the energy structure, enabling exact Gauss law preservation and energy behavior that shadows the continuous system. Numerical results in 1D and 2D demonstrate optimal convergence, Casimir conservation, and accurate representation of nonlinear phenomena such as higher-harmonic generation and traveling-wave solutions, with explicit treatment of dissipative effects and time-dependent sources. The framework provides a robust, extensible approach for time-domain simulations of a broad class of nonlinear optical problems, with potential extensions to anisotropic media and broken-FEEC/HDG variants.

Abstract

A simple Hamiltonian modeling framework for general models in nonlinear optics is given. This framework is specialized to describe the Hamiltonian structure of electromagnetic phenomena in cubicly nonlinear optical media. The model has a simple Poisson bracket structure with the Hamiltonian encoding all of the nonlinear coupling of the fields. The field-independence of the Poisson bracket facilitates a straightforward Hamiltonian structure-preserving discretization using finite element exterior calculus. The generality and relative simplicity of this Hamiltonian framework makes it amenable for simulating a broad class of time-domain nonlinear optical problems. The main contribution of this work is a finite element discretization of Maxwell's equations in cubicly nonlinear media which is energy-stable and exactly conserves Gauss's laws. Moreover, this approach may be readily adapted to consider more general nonlinear media in subsequent work.
Paper Structure (45 sections, 34 theorems, 263 equations, 15 figures)

This paper contains 45 sections, 34 theorems, 263 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Maxwell's equations in nonlinear media
  3. A general Hamiltonian model of Maxwell's equations in nonlinear media
  4. A Hamiltonian model of Maxwell's equations in cubicly nonlinear media
  5. Dissipative dynamics
  6. One- and two-dimensional models
  7. Model in dimensionless units
  8. A brief overview of FEEC methods and the discretization of functional derivatives
  9. FEEC method: A notational overview
  10. The dual de Rham complex
  11. Discrete functional derivatives
  12. Adjoint of degrees of freedom operator
  13. A Hamiltonian structure-preserving discretization of Maxwell's equations in general media
  14. An exterior calculus formulation of Maxwell's equations in nonlinear media
  15. A FEEC discretization of Maxwell's equations in nonlinear media
  16. ...and 30 more sections

Key Result

Theorem 1

Let $\overline{H}[\bm{D}, \bm{B}, \bm{P}, \bm{J}, Q, \sigma] = H[\bm{E}, \bm{B}, \bm{P}, \bm{J}, Q, \sigma]$ where $H$ is defined as in equation eq:cubic_3d_ham. Then the derivatives are given by

Figures (15)

  • Figure 1: Convergence rates of the relative error in the electric and magnetic fields in the $L^2(\Omega \times [0,T])$ and $L^\infty(\Omega \times [0,T])$ norms for the one-dimensional manufactured solution test. Optimal convergence is achieved except for the $E$-field when $4^{th}$-order polynomial interpolation is used.
  • Figure 2: Convergence rates of the relative error in the electric and magnetic fields in the $L^2(\Omega \times [0,T])$ and $L^\infty(\Omega \times [0,T])$ norms for the two-dimensional manufactured solution test.
  • Figure 3: Visualization of the evolution of the simple Fourier modes with dissipation. The top plot shows the solution in $t \in (0,10)$, the middle shows the solution in $t \in (90,100)$, and the bottom shows the solution in Fourier space.
  • Figure 4: Energy and Casimir conservation results for the simple Fourier mode test case with dissipation. The 1D discrete Casimir invariants are defined in equation \ref{['eq:1d_disc_casimir']}.
  • Figure 5: Energy and Casimir conservation results for the simple Fourier mode test case without dissipation. The 1D discrete Casimir invariants are defined in equation \ref{['eq:1d_disc_casimir']}. One can see that the energy is conserved on average in the dissipation free case.
  • ...and 10 more figures

Theorems & Definitions (36)

  • Theorem 1
  • Theorem 2
  • Definition 1
  • Definition 2
  • Proposition 1
  • Theorem 3
  • Lemma 1
  • Theorem 4
  • Theorem 5
  • Proposition 2
  • ...and 26 more