fdfd_tools/meanas/fdfd/functional.py

"""
Functional versions of many FDFD operators. These can be useful for performing
 FDFD calculations without needing to construct large matrices in memory.

The functions generated here expect `field_t` inputs with shape (3, X, Y, Z),
e.g. E = [E_x, E_y, E_z] where each component has shape (X, Y, Z)
"""
from typing import List, Callable, Tuple
import numpy

from .. import dx_lists_t, field_t
from ..fdmath.functional import curl_forward, curl_back

__author__ = 'Jan Petykiewicz'


field_transform_t = Callable[[field_t], field_t]


def e_full(omega: complex,
           dxes: dx_lists_t,
           epsilon: field_t,
           mu: field_t = None
           ) -> field_transform_t:
    """
    Wave operator for use with E-field. See `operators.e_full` for details.

    Args:
        omega: Angular frequency of the simulation
        dxes: Grid parameters [dx_e, dx_h] as described in meanas.types
        epsilon: Dielectric constant
        mu: Magnetic permeability (default 1 everywhere)

    Return:
        Function `f` implementing the wave operator
        `f(E)` -> `-i * omega * J`
    """
    ch = curl_back(dxes[1])
    ce = curl_forward(dxes[0])

    def op_1(e):
        curls = ch(ce(e))
        return curls - omega ** 2 * epsilon * e

    def op_mu(e):
        curls = ch(mu * ce(e))
        return curls - omega ** 2 * epsilon * e

    if numpy.any(numpy.equal(mu, None)):
        return op_1
    else:
        return op_mu


def eh_full(omega: complex,
            dxes: dx_lists_t,
            epsilon: field_t,
            mu: field_t = None
            ) -> Callable[[field_t, field_t], Tuple[field_t, field_t]]:
    """
    Wave operator for full (both E and H) field representation.
    See `operators.eh_full`.

    Args:
        omega: Angular frequency of the simulation
        dxes: Grid parameters [dx_e, dx_h] as described in meanas.types
        epsilon: Dielectric constant
        mu: Magnetic permeability (default 1 everywhere)

    Returns:
        Function `f` implementing the wave operator
        `f(E, H)` -> `(J, -M)`
    """
    ch = curl_back(dxes[1])
    ce = curl_forward(dxes[0])

    def op_1(e, h):
        return (ch(h) - 1j * omega * epsilon * e,
                ce(e) + 1j * omega * h)

    def op_mu(e, h):
        return (ch(h) - 1j * omega * epsilon * e,
                ce(e) + 1j * omega * mu * h)

    if numpy.any(numpy.equal(mu, None)):
        return op_1
    else:
        return op_mu


def e2h(omega: complex,
        dxes: dx_lists_t,
        mu: field_t = None,
        ) -> field_transform_t:
    """
    Utility operator for converting the `E` field into the `H` field.
    For use with `e_full` -- assumes that there is no magnetic current `M`.

    Args:
        omega: Angular frequency of the simulation
        dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.types`
        mu: Magnetic permeability (default 1 everywhere)

    Return:
        Function `f` for converting `E` to `H`,
        `f(E)` -> `H`
    """
    ce = curl_forward(dxes[0])

    def e2h_1_1(e):
        return ce(e) / (-1j * omega)

    def e2h_mu(e):
        return ce(e) / (-1j * omega * mu)

    if numpy.any(numpy.equal(mu, None)):
        return e2h_1_1
    else:
        return e2h_mu


def m2j(omega: complex,
        dxes: dx_lists_t,
        mu: field_t = None,
        ) -> field_transform_t:
    """
    Utility operator for converting magnetic current `M` distribution
    into equivalent electric current distribution `J`.
    For use with e.g. `e_full`.

    Args:
        omega: Angular frequency of the simulation
        dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.types`
        mu: Magnetic permeability (default 1 everywhere)

    Returns:
        Function `f` for converting `M` to `J`,
        `f(M)` -> `J`
    """
    ch = curl_back(dxes[1])

    def m2j_mu(m):
        J = ch(m / mu) / (-1j * omega)
        return J

    def m2j_1(m):
        J = ch(m) / (-1j * omega)
        return J

    if numpy.any(numpy.equal(mu, None)):
        return m2j_1
    else:
        return m2j_mu


def e_tfsf_source(TF_region: field_t,
                  omega: complex,
                  dxes: dx_lists_t,
                  epsilon: field_t,
                  mu: field_t = None,
                  ) -> field_transform_t:
    """
    Operator that turns an E-field distribution into a total-field/scattered-field
    (TFSF) source.

    Args:
        TF_region: mask which is set to 1 in the total-field region, and 0 elsewhere
                   (i.e. in the scattered-field region).
                   Should have the same shape as the simulation grid, e.g. `epsilon[0].shape`.
        omega: Angular frequency of the simulation
        dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.types`
        epsilon: Dielectric constant distribution
        mu: Magnetic permeability (default 1 everywhere)

    Returns:
        Function `f` which takes an E field and returns a current distribution,
        `f(E)` -> `J`
    """
    # TODO documentation
    A = e_full(omega, dxes, epsilon, mu)

    def op(e):
        neg_iwj = A(TF_region * e) - TF_region * A(e)
        return neg_iwj / (-1j * omega)


def poynting_e_cross_h(dxes: dx_lists_t) -> Callable[[field_t, field_t], field_t]:
    """
    Generates a function that takes the single-frequency `E` and `H` fields
    and calculates the cross product `E` x `H` = \\( E \\times H \\) as required
    for the Poynting vector, \\( S = E \\times H \\)

    Note:
        This function also shifts the input `E` field by one cell as required
        for computing the Poynting cross product (see `meanas.fdfd` module docs).

    Note:
        If `E` and `H` are peak amplitudes as assumed elsewhere in this code,
        the time-average of the poynting vector is `<S> = Re(S)/2 = Re(E x H) / 2`.
        The factor of `1/2` can be omitted if root-mean-square quantities are used
        instead.

    Args:
        dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.types`

    Returns:
        Function `f` that returns E x H as required for the poynting vector.
    """
    def exh(e: field_t, h: field_t):
        s = numpy.empty_like(e)
        ex = e[0] * dxes[0][0][:, None, None]
        ey = e[1] * dxes[0][1][None, :, None]
        ez = e[2] * dxes[0][2][None, None, :]
        hx = h[0] * dxes[1][0][:, None, None]
        hy = h[1] * dxes[1][1][None, :, None]
        hz = h[2] * dxes[1][2][None, None, :]
        s[0] = numpy.roll(ey, -1, axis=0) * hz - numpy.roll(ez, -1, axis=0) * hy
        s[1] = numpy.roll(ez, -1, axis=1) * hx - numpy.roll(ex, -1, axis=1) * hz
        s[2] = numpy.roll(ex, -1, axis=2) * hy - numpy.roll(ey, -1, axis=2) * hx
        return s
    return exh