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FMMAX: Fourier Modal Method with Jax

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FMMAX is a an implementation of the Fourier modal method (FMM) in JAX.

The FMM -- also known as rigorous coupled wave analysis (RCWA) -- is a semianalytical method that solves Maxwell's equations in periodic stratified media, where in-plane directions are treated with a truncated Fourier basis and the normal direction is handled by a scattering matrix approach [1999 Whittaker, 2012 Liu, 2020 Jin]. This allows certain classes of structures to be modeled with relatively low computational cost.

Our use of JAX enables GPU acceleration and automatic differentiation of FMM simulations. Besides these features, FMMAX is differentiated from other codes by its support for Brillouin zone integration, advanced vector FMM formulations which improve convergence, and anisotropic and magnetic materials.

Brillouin zone integration

Brillouin zone integration [2022 Lopez-Fraguas] allows modeling of localized sources in periodic structures. Check out the crystal example to see how we model a Gaussian beam incident upon a photonic crystal slab, or an isolated dipole embedded within the slab. The Gaussian beam fields are shown below.

Gaussian beam incident on photonic crystal

Vector FMM formulations

Vector FMM formulations introduce local coordinate systems at each point in the unit cell, which are normal and tangent to all interfaces. This allows normal and tangent field components to be treated differently and improves convergence. FMMAX implements several vector formulations of the FMM, with automatic vector field generation based on functional minimization similar to [2012 Liu]. We implement the Pol, Normal, and Jones methods of that reference, and introduce a new Jones direct method which we have found to have superior convergence. These are supported also with anisotropic and magnetic materials. The vector_fields example computes vector fields by these methods for an example structure.

Comparison of automatically-generated vector fields

Anisotropic, magnetic materials

Our support of anisotropic, magnetic materials allows modeling of uniaxial perfectly matched layers. This is demonstrated in the metal_dipole example, which simulates in vaccuum located above a metal substrate. The resulting electric fields are whown below.

Dipole suspended above metal substrate with PML

FMM Conventions

Batching

Batched calculations are supported, and should be used where possible to avoid looping. The batch axes are the leading axes, except for the wave amplitudes and electromagnetic fields, where a trailing batch axis is assumed. This allows e.g. computing the transmission through a structure for multiple polarizations via a matrix-matrix operation (transmitted_amplitudes = S11 @ incident_amplitudes), rather than a batched matrix-vector operation.

Installation

FMMAX can be installed via pip:

pip install fmmax

For developers requiring a local installation, you will need to first clone this repository and then perform a local install from within the root directory using:

pip install -e ".[dev]"

The [dev] modifier specifies optional dependencies for developers which are listed in pyproject.toml.

Note: for this to work, it may be necessary to first update your pip installation using e.g. python3 -m pip install --upgrade pip.

Citing FMMAX

If you use FMMAX, please consider citing our paper,

@misc{schubert2023fourier,
      title={Fourier modal method for inverse design of metasurface-enhanced micro-LEDs}, 
      author={Martin F. Schubert and Alec M. Hammond},
      year={2023},
      eprint={2308.08573},
      archivePrefix={arXiv},
      primaryClass={physics.comp-ph}
}

License

FMMAX is licensed under the MIT license.

References