Unit cell and basis

Unit cell and basis#

fmmax.X = Array([1., 0.], dtype=float32)#

Unit vector pointing in the x direction.

fmmax.Y = Array([0., 1.], dtype=float32)#

Unit vector pointing in the y direction.

class fmmax.Expansion(basis_coefficients: ndarray[Any, Any])[source]#

Stores the expansion.

The expansion consists of the integer coefficients of the reciprocal lattice vectors used in the Fourier expansion of fields, permittivities, etc. in the FMM scheme.

basis_coefficients#

The integer coefficients of the primitive reciprocal lattice vectors, which generate the full set of reciprocal-space vectors in the expansion.

Type:

numpy.ndarray[Any, Any]

num_terms#

The number of terms in the expansion.

class fmmax.LatticeVectors(u: Array, v: Array)[source]#

Stores a pair of lattice vectors.

Note that this is just a pair of 2-dimensional vectors, which may be either for the real-space lattice or the reciprocal space lattice, depending on usage.

u#

The first primitive lattice vector.

Type:

jax.Array

v#

The second primitive lattice vector, with identical shape.

Type:

jax.Array

class fmmax.Truncation(value)[source]#

Enumerates truncation modes.

CIRCULAR = 'circular'#

Fourier orders are truncated based on the total magnitude of their associated wavevectors.

PARALLELOGRAMIC = 'parallelogramic'#

Fourier orders are truncated based on the magnitude of their component along u and v directions, without regard for their total magnitude.

fmmax.brillouin_zone_in_plane_wavevector(brillouin_grid_shape: Tuple[int, int], primitive_lattice_vectors: LatticeVectors) → Array[source]#

Computes in-plane wavevectors on a regular grid in the first Brillouin zone.

The wavevectors are found by dividing the Brillouin zone into a grid with the specified shape; the wavevectors are at the centers of the grid voxels.

Parameters:

  • brillouin_grid_shape – The shape of the wavevector grid.

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

Returns:

The in-plane wavevectors, with shape brillouin_grid_shape + (2,).

fmmax.generate_expansion(primitive_lattice_vectors: LatticeVectors, approximate_num_terms: int, truncation: Truncation = Truncation.CIRCULAR) → Expansion[source]#

Generates the expansion for the specified real-space basis.

Parameters:

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

  • approximate_num_terms – The approximate number of terms in the expansion. To maintain a symmetric expansion, the total number of terms may differ from this value.

  • truncation – The truncation to be used for the expansion. The default is Truncation.CIRCULAR.

Returns:

The Expansion. The basis coefficients of the expansion are sorted so that the zeroth-order term is first.

fmmax.min_array_shape_for_expansion(expansion: Expansion) → Tuple[int, int][source]#

Returns the minimum allowed shape compatible with expansion.

fmmax.plane_wave_in_plane_wavevector(wavelength: Array, polar_angle: Array, azimuthal_angle: Array, permittivity: Array) → Array[source]#

Computes the in-plane wavevector for a plane-wave excitation.

Parameters:

  • wavelength – The free-space wavelength of the plane-wave excitation.

  • polar_angle – Polar angle of the plane-wave excitation.

  • azimuthal_angle – Azimuthal angle of plane-wave the excitation.

  • permittivity – Scalar permittivity of the medium in which the polar angle and azimuthal angle are specified.

Returns:

The fundamental transverse wavevector, i.e. (kx0, ky0).

fmmax.transverse_wavevectors(in_plane_wavevector: Array, primitive_lattice_vectors: LatticeVectors, expansion: Expansion) → Array[source]#

Obtains all the relevant transverse wavevectors given the expansion.

Parameters:

  • in_plane_wavevector – The zeroth-order transverse wavevector.

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

  • expansion – The expansion for which the set of transverse wavevectors are sought.

Returns:

The transverse wavevectors.

fmmax.unit_cell_coordinates(primitive_lattice_vectors: LatticeVectors, shape: Tuple[int, int], num_unit_cells: Tuple[int, int] = (1, 1)) → Tuple[Array, Array][source]#

Compute spatial coordinates given the grid shape and number of unit cells.

Parameters:

  • primitive_lattice_vectors – The lattice vectors defining te unit cell.

  • shape – The shape of the coordinates grid.

  • num_unit_cells – Determines the number of unit cells for which to compute the coordinates. Default is (1, 1), corresponding to a single unit cell.

Returns:

The unit cell coordinates, with shape (shape[0] * num_unit_cells[0], shape[1] * num_unit_cells[1]).

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