7th Asia-Paci c Conference on Plasma Physics, 12-17 Nov, 2023 at Port Messe Nagoya

A Hamiltonian model and its structure preserving discretization of theponderomotive force in 1D
William Barham1,2, Yaman Güçlü4, Philip J. Morrison2,3, and Eric Sonnendrücker4,5
1 Oden Institute for Computational Science Engineering and Sciences, University of Texas Austin
2 Institute for Fusion Studies, University of Texas Austin
3 Department of Physics, University of Texas Austin
4 Max Planck Institute for Plasma Physics, Numerical Methods in Plasma Physics
5 Department of Mathematics, Technical University of Muniche-mail (speaker): william.barham@utexas.edu

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A large number of interactions between matter and theelectromagnetic field may be adequately captured byapproximate models which reduce the interaction to anonlinear polarization of the medium. In many ways, thisconcept is the heart of nonlinear optics, but also bears
relevance in plasma physics in which asymptotic
approximations often induce field dependent polarizationand magnetization of the electromagnetic fields: see e.g.
[1] for nonlinear electromagnetic media arising fromlifting particle-orbit asymptotics to kinetic theories. In
this work, we derive a general Hamiltonian structure
preserving discretization of Maxwell's equations in
nonlinear media which might be applied to broad class ofmodels, derive a self-consistent Hamiltonian model ofthe ponderomotive force to which we might apply thisgeneral discretization procedure, and finally investigate
the qualities of the general method as applied to thisparticular system.
A well known laser-plasma interaction model gives
rise to a nonlinear polarization of the electromagnetic
medium. In the presence of an inhomogeneous
oscillatory electric field, charged particles experience anet force, averaged over the oscillatory timescale, known
as the ponderomotive force. Starting from the two-fluid
equations, we derive a Hamiltonian model in which theponderomotive force provides a current source forMaxwell's equations. The asymptotic assumptions underwhich this model is derived correspond to what isfrequently called the “linear regime” in laser-plasmainteraction models [2]. Our model is possibly thesimplest self-consistent model of the ponderomotive
force, and closely resembles the model studied in [3].
Moreover, our model is perhaps the first to elucidate theHamiltonian structure of the 1D linear regime, andprovides an alternative approach to its derivation basedon applying the asymptotics directly to the energyfunctional and Poisson bracket rather than the equationsof motion. We call the system of equations so derived the
1D ponderomotive Maxwell system.
One finds a simple Poisson bracket for the 1D
ponderomotive Maxwell system: it is a direct sum of thePoisson brackets for Maxwell and acoustic wave
equations, as well as a coupling bracket. Because all ofthese brackets are field-free, a structure preserving
discretization is easily accomplished using finite element
exterior calculus (FEEC). In particular, we use a spectral
element FEEC method. We explore the behavior ofdiscretizations based on not only a traditional spectral
element FEEC approach, but also a discontinuous
broken-FEEC approach, see [4,5]. Temporal

discretization is accomplished using Hamiltoniansplitting.
Using both conforming and broken-FEEC methods
with both low and high order discretizations in space andtime, we simulate the 1D ponderomotive Maxwell
system in order to better understand the properties of thisnew approach to discretizing Maxwell's equations innonlinear media. In all cases, we find good conservationof energy to the order of the splitting method used, andconservation Casimir invariants (e.g. Gauss's laws) tomachine precision. To our knowledge this is the firsttime-domain solver for Maxwell's equations in nonlinear
media which conserves energy and Gauss's laws,
accommodates high order discretizations in both space
and time, and which may use local, discontinuous basis
functions to localize nonlinear solves (allowing forparallel implementation). The methods used to discretize
the 1D ponderomotive Maxwell system shows promise
to generalize well to a broad class of models ofelectrodynamics in nonlinear media. Hence, thesefavorable qualities found in method used to study the 1D
ponderomotive Maxwell system might be used to study amuch broader class of models in nonlinear optics andplasma physics.
References[1] P. J. Morrison, “A general theory for gauge-freelifting,” Physics of Plasmas 20, 012104 (2013).
[2] E. Esarey, P. Sprangle, J. Krall, and A. Ting, “Selffocusing and guiding of short laser pulses in ionizing
gases and plasmas,” IEEE Journal of Quantum
Electronics 33, 1879–1914 (1997).
[3] J. Antonsen, T. M. and P. Mora, “Self-focusing andRaman scattering of laser pulses in tenuous plasmas,”
Physics of Fluids B: Plasma Physics 5, 1440–1452(1993).
[4] M. Campos-Pinto and Y. Güçlü, “Broken-FEEC
approximations of Hodge Laplace problems,” (2021).
[5] Y. Güçlü, S. Hadjout, and M. C. Pinto, “A broken
feec framework for electromagnetic problems on mapped
multipatch domains,” (2022), arXiv:2208.05238.