A Hamiltonian structure-preserving discretization of Maxwell's equations in nonlinear media
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866917771240013824 |
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| author | Barham, William Güçlü, Yaman Morrison, Philip J. Sonnendrücker, Eric |
| author_facet | Barham, William Güçlü, Yaman Morrison, Philip J. Sonnendrücker, Eric |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_04891 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Hamiltonian structure-preserving discretization of Maxwell's equations in nonlinear media Barham, William Güçlü, Yaman Morrison, Philip J. Sonnendrücker, Eric Computational Physics Numerical Analysis 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. |
| title | A Hamiltonian structure-preserving discretization of Maxwell's equations in nonlinear media |
| topic | Computational Physics Numerical Analysis |
| url | https://arxiv.org/abs/2409.04891 |