fmm

fmm#

Functions related to layer eigenmode calculation for the FMM algorithm.

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class fmmax.fmm.Formulation(value)[source]#

Enumerates supported Fourier modal method formulations.

class fmmax.fmm.LayerSolveResult(wavelength: Array, in_plane_wavevector: Array, primitive_lattice_vectors: LatticeVectors, expansion: Expansion, eigenvalues: Array, eigenvectors: Array, z_permittivity_matrix: Array, inverse_z_permittivity_matrix: Array, z_permeability_matrix: Array, inverse_z_permeability_matrix: Array, transverse_permeability_matrix: Array, tangent_vector_field: Tuple[Array, Array] | None)[source]#

Stores the result of a layer eigensolve.

This eigenvalue problem is specified in equation 28 of [2012 Liu].

wavelength#

The wavelength for the solve.

Type:

jax.Array

in_plane_wavevector#

The in-plane wavevector for the solve.

Type:

jax.Array

primitive_lattice_vectors#

The primitive vectors for the real-space lattice.

Type:

fmmax.basis.LatticeVectors

expansion#

The expansion used for the eigensolve.

Type:

fmmax.basis.Expansion

eigenvalues#

The layer eigenvalues.

Type:

jax.Array

eigenvectors#

The layer eigenvectors.

Type:

jax.Array

z_permittivity_matrix#

The fourier-transformed zz-component of permittivity.

Type:

jax.Array

inverse_z_permittivity_matrix#

The fourier-transformed inverse of zz-component of permittivity.

Type:

jax.Array

z_permeability_matrix#

The fourier-transformed zz-component of permeability.

Type:

jax.Array

inverse_z_permeability_matrix#

The fourier-transformed inverse of zz-component of permeability.

Type:

jax.Array

transverse_permeability_matrix#

The transverse permeability matrix, needed to calculate the omega-script-k matrix from equation 26 of [2012 Liu]. This is needed to generate the layer scattering matrix.

Type:

jax.Array

tangent_vector_field#

The tangent vector field (tx, ty) used to compute the transverse permittivity matrix, if a vector FMM formulation is used. If the FFT formulation is used, the vector field is None.

Type:

Tuple[jax.Array, jax.Array] | None

property omega_script_k_matrix#

Compute omega-script-k matrix from equation 26 of [2012 Liu].

fmmax.fmm.eigensolve_anisotropic_media(wavelength: Array, in_plane_wavevector: Array, primitive_lattice_vectors: LatticeVectors, permittivity_xx: Array, permittivity_xy: Array, permittivity_yx: Array, permittivity_yy: Array, permittivity_zz: Array, expansion: Expansion, formulation: Formulation | Callable[[Array, Expansion, LatticeVectors], Tuple[Array, Array]]) → LayerSolveResult[source]#

Performs the eigensolve for a layer with anisotropic permittivity.

This function performs either a uniform-layer or patterned-layer eigensolve, depending on the shape of the trailing dimensions of a given layer permittivity. When the final two dimensions have shape (1, 1), the layer is treated as uniform. Otherwise, it is patterned.

Parameters:

  • wavelength – The free space wavelength of the excitation.

  • in_plane_wavevector – (kx0, ky0).

  • primitive_lattice_vectors – The primitive vectors for the real-space lattice.

  • permittivity_xx – The xx-component of the permittivity tensor, with shape (…, nx, ny).

  • permittivity_xy – The xy-component of the permittivity tensor.

  • permittivity_yx – The yx-component of the permittivity tensor.

  • permittivity_yy – The yy-component of the permittivity tensor.

  • permittivity_zz – The zz-component of the permittivity tensor.

  • expansion – The field expansion to be used.

  • formulation – Specifies the formulation to be used, or a callable which computes the tangent vector field for a custom vector FMM formulation.

Returns:

The LayerSolveResult.

fmmax.fmm.eigensolve_general_anisotropic_media(wavelength: Array, in_plane_wavevector: Array, primitive_lattice_vectors: LatticeVectors, permittivity_xx: Array, permittivity_xy: Array, permittivity_yx: Array, permittivity_yy: Array, permittivity_zz: Array, permeability_xx: Array, permeability_xy: Array, permeability_yx: Array, permeability_yy: Array, permeability_zz: Array, expansion: Expansion, formulation: Formulation | Callable[[Array, Expansion, LatticeVectors], Tuple[Array, Array]], vector_field_source: Array | None = None) → LayerSolveResult[source]#

Performs the eigensolve for a general anistropic layer.

Here, “general” refers to the fact that the layer material can be magnetic, i.e. the permeability and permittivity can be specified.

This function performs either a uniform-layer or patterned-layer eigensolve, depending on the shape of the trailing dimensions of a given layer permittivity. When the final two dimensions have shape (1, 1), the layer is treated as uniform. Otherwise, it is patterned.

Parameters:

  • wavelength – The free space wavelength of the excitation.

  • in_plane_wavevector – (kx0, ky0).

  • primitive_lattice_vectors – The primitive vectors for the real-space lattice.

  • permittivity_xx – The xx-component of the permittivity tensor, with shape (…, nx, ny).

  • permittivity_xy – The xy-component of the permittivity tensor.

  • permittivity_yx – The yx-component of the permittivity tensor.

  • permittivity_yy – The yy-component of the permittivity tensor.

  • permittivity_zz – The zz-component of the permittivity tensor.

  • permeability_xx – The xx-component of the permeability tensor.

  • permeability_xy – The xy-component of the permeability tensor.

  • permeability_yx – The yx-component of the permeability tensor.

  • permeability_yy – The yy-component of the permeability tensor.

  • permeability_zz – The zz-component of the permeability tensor.

  • expansion – The field expansion to be used.

  • formulation – Specifies the formulation to be used, or a callable which computes the tangent vector field for a custom vector FMM formulation.

  • vector_field_source – Optional array used to calculate the vector field for vector formulations of the FMM. If not specified, (permittivity_xx + permittivity_yy) / 2 is used. Ignored for the FFT formulation. Should have shape matching the permittivities and permeabilities.

Returns:

The LayerSolveResult.

fmmax.fmm.eigensolve_isotropic_media(wavelength: Array, in_plane_wavevector: Array, primitive_lattice_vectors: LatticeVectors, permittivity: Array, expansion: Expansion, formulation: Formulation | Callable[[Array, Expansion, LatticeVectors], Tuple[Array, Array]]) → LayerSolveResult[source]#

Performs the eigensolve for a layer with isotropic permittivity.

This function performs either a uniform-layer or patterned-layer eigensolve, depending on the shape of the trailing dimensions of a given layer permittivity. When the final two dimensions have shape (1, 1), the layer is treated as uniform. Otherwise, it is patterned.

Parameters:

  • wavelength – The free space wavelength of the excitation.

  • in_plane_wavevector – (kx0, ky0).

  • primitive_lattice_vectors – The primitive vectors for the real-space lattice.

  • permittivity – The permittivity array.

  • expansion – The field expansion to be used.

  • formulation – Specifies the formulation to be used, or a callable which computes the tangent vector field for a custom vector FMM formulation.

Returns:

The LayerSolveResult.