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Cpu Numba

particle_interaction

Calculates the interaction between particles based on their properties and returns the result.

Parameters:

Name Type Description Default
lmax int

The maximum value of the angular momentum quantum number l. It determines the size of the arrays plm and sph_h.

 required
particle_number int

The number of particles in the system.

 required
idx ndarray

A numpy array of shape (jmax, 5), where jmax is the total number of interactions between particles. Each row of idx represents an interaction and contains the following information: - s1 (int): The index of the first particle. - n1 (int): The index of the first particle's property. - tau1 (int): The tau value of the first particle. - l1 (int): The l value of the first particle. - m1 (int): The m value of the first particle.

 required
x ndarray

A numpy array representing the positions of the particles. It has shape (particle_number,) and contains the x-coordinates of the particles.

 required
translation_table ndarray

A 3-dimensional numpy array that stores the translation coefficients used in the calculation. It has shape (n2, n1, p) where n2 and n1 are the indices of the translation coefficients, and p is the maximum.

 required
plm ndarray

A numpy array representing the associated Legendre polynomials. It has shape (pmax * (pmax + 1) // 2, s1max, s2max), where pmax is the maximum degree of the Legendre polynomials.

 required
sph_h ndarray

A numpy array representing the spherical harmonics. It has shape (jmax, jmax, channels), where jmax is the maximum number of particles, channels is the number of channels.

 required
e_j_dm_phi ndarray

The parameter e_j_dm_phi is not defined in the code snippet you provided. Could you please provide the definition or explanation of what e_j_dm_phi represents?

 required

Returns:

Name Type Description
wx ndarray

The array wx, which represents the result of the particle interaction calculations.
Source code in yasfpy/functions/cpu_numba.py 
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@jit(nopython=True, parallel=True, nogil=True, fastmath=True, cache=True)
def particle_interaction(
    lmax: int,
    particle_number: int,
    idx: np.ndarray,
    x: np.ndarray,
    translation_table: np.ndarray,
    plm: np.ndarray,
    sph_h: np.ndarray,
    e_j_dm_phi,
):
    """Calculates the interaction between particles based on their properties and returns the result.

    Args:
        lmax (int): The maximum value of the angular momentum quantum number `l`. It determines the size of the arrays `plm` and `sph_h`.
        particle_number (int): The number of particles in the system.
        idx (np.ndarray): A numpy array of shape `(jmax, 5)`, where `jmax` is the total number of interactions between particles. Each row of `idx` represents an interaction and contains the following information:
            - s1 (int): The index of the first particle.
            - n1 (int): The index of the first particle's property.
            - tau1 (int): The tau value of the first particle.
            - l1 (int): The l value of the first particle.
            - m1 (int): The m value of the first particle.
        x (np.ndarray): A numpy array representing the positions of the particles. It has shape `(particle_number,)` and contains the x-coordinates of the particles.
        translation_table (np.ndarray): A 3-dimensional numpy array that stores the translation coefficients used in the calculation. It has shape `(n2, n1, p)` where `n2` and `n1` are the indices of the translation coefficients, and `p` is the maximum.
        plm (np.ndarray): A numpy array representing the associated Legendre polynomials. It has shape `(pmax * (pmax + 1) // 2, s1max, s2max)`, where `pmax` is the maximum degree of the Legendre polynomials.
        sph_h (np.ndarray): A numpy array representing the spherical harmonics. It has shape `(jmax, jmax, channels)`, where `jmax` is the maximum number of particles, `channels` is the number of channels.
        e_j_dm_phi (np.ndarray): The parameter `e_j_dm_phi` is not defined in the code snippet you provided. Could you please provide the definition or explanation of what `e_j_dm_phi` represents?

    Returns:
        wx (np.ndarray): The array `wx`, which represents the result of the particle interaction calculations.
    """
    jmax = particle_number * 2 * lmax * (lmax + 2)
    channels = sph_h.shape[-1]

    wx = np.zeros(x.size * channels, dtype=complex128).reshape(x.shape + (channels,))

    for w_idx in prange(jmax * jmax * channels):
        w = w_idx % channels
        j_idx = w_idx // channels
        j1 = j_idx // jmax
        j2 = j_idx % jmax
        s1, n1, tau1, l1, m1 = idx[j1, :]
        s2, n2, tau2, l2, m2 = idx[j2, :]

        if s1 == s2:
            continue

        delta_tau = np.absolute(tau1 - tau2)
        delta_l = np.absolute(l1 - l2)
        delta_m = np.absolute(m1 - m2)

        val = 0j
        for p in range(np.maximum(delta_m, delta_l + delta_tau), l1 + l2 + 1):
            val += (
                translation_table[n2, n1, p]
                * plm[p * (p + 1) // 2 + delta_m, s1, s2]
                * sph_h[p, s1, s2, w]
            )
        val *= e_j_dm_phi[m2 - m1 + 2 * lmax, s1, s2] * x[j2]

        wx[j1, w] += val

    return wx

compute_idx_lookups

The function compute_idx_lookups generates an index lookup table for a given lmax and particle_number using parallel processing.

Parameters:

Name Type Description Default
lmax int

The maximum value of the angular momentum quantum number l. It determines the range of values for l in the nested loop.

 required
particle_number int

The number of particles in the system.

 required

Returns:

Name Type Description
idx ndarray

A NumPy array idx which contains the computed index lookups.
Source code in yasfpy/functions/cpu_numba.py 
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@jit(nopython=True, parallel=True, fastmath=True)
def compute_idx_lookups(lmax: int, particle_number: int):
    """
    The function `compute_idx_lookups` generates an index lookup table for a given `lmax` and
    `particle_number` using parallel processing.

    Args:
        lmax (int): The maximum value of the angular momentum quantum number `l`. It determines the range of values for `l` in the nested loop.
        particle_number (int): The number of particles in the system.

    Returns:
        idx (np.ndarray): A NumPy array `idx` which contains the computed index lookups.
    """
    nmax = 2 * lmax * (lmax + 2)
    idx = np.zeros(nmax * particle_number * 5, dtype=int64).reshape(
        (nmax * particle_number, 5)
    )

    for s in prange(particle_number):
        for tau in range(1, 3):
            for l in range(1, lmax + 1):
                for m in range(-l, l + 1):
                    n = (tau - 1) * lmax * (lmax + 2) + (l - 1) * (l + 1) + l + m
                    i = n + s * nmax
                    idx[i, 0] = s
                    idx[i, 1] = n
                    idx[i, 2] = tau
                    idx[i, 3] = l
                    idx[i, 4] = m

    return idx

compute_scattering_cross_section

Calculates the scattering cross section for a given set of input parameters.

Parameters:

Name Type Description Default
lmax int

The maximum angular momentum quantum number. It determines the maximum value of l in the calculations.

 required
particle_number int

The number of particles in the system.

 required
idx ndarray

A numpy array of shape (jmax, 5), where jmax is the total number of particle pairs. Each row of idx represents a particle pair and contains the following information: - s (int): The index of the first particle. - n (int): The index of the second particle. - tau (int): The tau value. - l (int): The l value. - m (int): The m value.

 required
sfc ndarray

A numpy array of shape (s, n, channels), where:

 required
translation_table ndarray

A 3-dimensional numpy array that stores the translation coefficients used in the computation of the scattering cross section. It has shape (n2, n1, p) where n2 and n1 are the number of radial functions for the second and first particles, respectively, and p is the order of the Legendre polynomial.

 required
plm ndarray

A numpy array representing the associated Legendre polynomials. It has shape (pmax * (pmax + 1) // 2, 2, 2), where pmax is the maximum value of p in the loop.

 required
sph_h ndarray

A numpy array of shape (pmax, s1max, s2max, channels). It represents the scattering matrix elements for each combination of s1, s2, and p, where p is the order of the Legendre polynomial.

 required
e_j_dm_phi ndarray

A numpy array representing the scattering phase function. It has shape (2*lmax+1, channels, channels) and contains complex values. The indices (j, s1, s2) represent the angular momentum index j, and the spin indices s1 and s2.

 required

Returns:

Name Type Description
c_sca_complex ndarray

The complex scattering cross section c_sca_complex.
Source code in yasfpy/functions/cpu_numba.py 
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@jit(nopython=True, parallel=True, nogil=True, fastmath=True)
def compute_scattering_cross_section(
    lmax: int,
    particle_number: int,
    idx: np.ndarray,
    sfc: np.ndarray,
    translation_table: np.ndarray,
    plm: np.ndarray,
    sph_h: np.ndarray,
    e_j_dm_phi: np.ndarray,
):
    """Calculates the scattering cross section for a given set of input parameters.

    Args:
        lmax (int): The maximum angular momentum quantum number. It determines the maximum value of `l` in the calculations.
        particle_number (int): The number of particles in the system.
        idx (np.ndarray): A numpy array of shape `(jmax, 5)`, where `jmax` is the total number of particle pairs. Each row of `idx` represents a particle pair and contains the following information:
            - s (int): The index of the first particle.
            - n (int): The index of the second particle.
            - tau (int): The tau value.
            - l (int): The l value.
            - m (int): The m value.
        sfc (np.ndarray): A numpy array of shape `(s, n, channels)`, where:
        translation_table (np.ndarray): A 3-dimensional numpy array that stores the translation coefficients used in the computation of the scattering cross section. It has shape `(n2, n1, p)` where `n2` and `n1` are the number of radial functions for the second and first particles, respectively, and `p` is the order of the Legendre polynomial.
        plm (np.ndarray): A numpy array representing the associated Legendre polynomials. It has shape `(pmax * (pmax + 1) // 2, 2, 2)`, where `pmax` is the maximum value of `p` in the loop.
        sph_h (np.ndarray): A numpy array of shape `(pmax, s1max, s2max, channels)`. It represents the scattering matrix elements for each combination of `s1`, `s2`, and `p`, where `p` is the order of the Legendre polynomial.
        e_j_dm_phi (np.ndarray): A numpy array representing the scattering phase function. It has shape `(2*lmax+1, channels, channels)` and contains complex values. The indices `(j, s1, s2)` represent the angular momentum index `j`, and the spin indices `s1` and `s2`.

    Returns:
        c_sca_complex (np.ndarray): The complex scattering cross section `c_sca_complex`.
    """
    jmax = particle_number * 2 * lmax * (lmax + 2)
    channels = sph_h.shape[-1]

    c_sca_complex = np.zeros(channels, dtype=complex128)

    for j_idx in prange(jmax * jmax):
        j1 = j_idx // jmax
        j2 = j_idx % jmax
        s1, n1, _, _, m1 = idx[j1, :]
        s2, n2, _, _, m2 = idx[j2, :]

        delta_m = np.absolute(m1 - m2)

        p_dependent = np.zeros(channels, dtype=complex128)
        for p in range(delta_m, 2 * lmax + 1):
            p_dependent += (
                translation_table[n2, n1, p]
                * plm[p * (p + 1) // 2 + delta_m, s1, s2]
                * sph_h[p, s1, s2, :]
            )
        p_dependent *= (
            np.conj(sfc[s1, n1, :])
            * e_j_dm_phi[m2 - m1 + 2 * lmax, s1, s2]
            * sfc[s2, n2, :]
        )

        c_sca_complex += p_dependent

    return c_sca_complex

compute_radial_independent_scattered_field_legacy

Calculates the scattered field for a given set of parameters and returns the result.

Parameters:

Name Type Description Default
lmax int

The maximum value of the angular momentum quantum number l. It determines the maximum order of the spherical harmonics used in the computation.

 required
particles_position ndarray

An array representing the positions of particles. It has shape (num_particles, 3), where num_particles is the number of particles and each row represents the x, y, and z coordinates of a particle.

 required
idx ndarray

An array containing the indices of the particles. It has shape (jmax, 5) where jmax is the total number of particles. Each row of idx represents a particle and contains the following information:

 required
sfc ndarray

A 3-dimensional array representing the scattering form factors. It has dimensions (s, n, w), where:

 required
k_medium ndarray

An array representing the wave number in the medium. It is used in the calculation of the scattered field.

 required
azimuthal_angles ndarray

An array of azimuthal angles, representing the angles at which the scattered field is computed.

 required
e_r ndarray

An array representing the radial component of the electric field. It has shape (azimuthal_angles.size, 3), where azimuthal_angles.size is the number of azimuthal angles and 3 represents the three Cartesian components of the electric field.

 required
e_phi ndarray

An array representing the electric field component in the azimuthal direction. It has a shape of (azimuthal_angles.size, 3), where azimuthal_angles.size is the number of azimuthal angles and 3 represents the three components of the electric field.

 required
e_theta ndarray

An array representing the electric field component in the theta direction. It has a shape of (azimuthal_angles.size, 3), where azimuthal_angles.size is the number of azimuthal angles and 3 represents the three components of the electric field.

 required
pilm ndarray

An array representing the matrix of spherical harmonics coefficients. It has a shape of (lmax+1, lmax+1, azimuthal_angles.size). Each element pilm[l, m, a] represents the coefficient of the spherical harmonics for a given l, m, and azimuthal angle a.

 required
taulm ndarray

An array representing the scattering coefficients for each combination of l, m, and azimuthal angle a. It has a shape of (lmax+1, lmax+1, azimuthal_angles.size). The values in taulm represent the scattering coefficients.

 required

Returns:

Name Type Description
e_1_sca ndarray

An array of complex numbers representing the scattered field.
Source code in yasfpy/functions/cpu_numba.py 
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@jit(nopython=True, parallel=True, nogil=True, fastmath=True)
def compute_radial_independent_scattered_field_legacy(
    lmax: int,
    particles_position: np.ndarray,
    idx: np.ndarray,
    sfc: np.ndarray,
    k_medium: np.ndarray,
    azimuthal_angles: np.ndarray,
    e_r: np.ndarray,
    e_phi: np.ndarray,
    e_theta: np.ndarray,
    pilm: np.ndarray,
    taulm: np.ndarray,
):
    """Calculates the scattered field for a given set of parameters and returns the result.

    Args:
        lmax (int): The maximum value of the angular momentum quantum number `l`. It determines the maximum order of the spherical harmonics used in the computation.
        particles_position (np.ndarray): An array representing the positions of particles. It has shape `(num_particles, 3)`, where `num_particles` is the number of particles and each row represents the x, y, and z coordinates of a particle.
        idx (np.ndarray): An array containing the indices of the particles. It has shape `(jmax, 5)` where `jmax` is the total number of particles. Each row of `idx` represents a particle and contains the following information:
        sfc (np.ndarray): A 3-dimensional array representing the scattering form factors. It has dimensions `(s, n, w)`, where:
        k_medium (np.ndarray): An array representing the wave number in the medium. It is used in the calculation of the scattered field.
        azimuthal_angles (np.ndarray): An array of azimuthal angles, representing the angles at which the scattered field is computed.
        e_r (np.ndarray): An array representing the radial component of the electric field. It has shape `(azimuthal_angles.size, 3)`, where `azimuthal_angles.size` is the number of azimuthal angles and 3 represents the three Cartesian components of the electric field.
        e_phi (np.ndarray): An array representing the electric field component in the azimuthal direction. It has a shape of `(azimuthal_angles.size, 3)`, where `azimuthal_angles.size` is the number of azimuthal angles and `3` represents the three components of the electric field.
        e_theta (np.ndarray): An array representing the electric field component in the theta direction. It has a shape of `(azimuthal_angles.size, 3)`, where `azimuthal_angles.size` is the number of azimuthal angles and `3` represents the three components of the electric field.
        pilm (np.ndarray): An array representing the matrix of spherical harmonics coefficients. It has a shape of `(lmax+1, lmax+1, azimuthal_angles.size)`. Each element `pilm[l, m, a]` represents the coefficient of the spherical harmonics for a given `l`, `m`, and azimuthal angle `a`.
        taulm (np.ndarray): An array representing the scattering coefficients for each combination of `l`, `m`, and azimuthal angle `a`. It has a shape of `(lmax+1, lmax+1, azimuthal_angles.size)`. The values in `taulm` represent the scattering coefficients.

    Returns:
        e_1_sca (np.ndarray): An array of complex numbers representing the scattered field.

    """
    e_1_sca = np.zeros(
        azimuthal_angles.size * 3 * k_medium.size, dtype=complex128
    ).reshape((azimuthal_angles.size, 3, k_medium.size))
    jmax = particles_position.shape[0] * 2 * lmax * (lmax + 2)

    for global_idx in prange(jmax * azimuthal_angles.size * k_medium.size):
        w_idx = global_idx % (jmax * k_medium.size)
        g_idx = global_idx // (jmax * k_medium.size)

        a = g_idx

        w = w_idx % k_medium.size
        j_idx = w_idx // k_medium.size
        s, n, tau, l, m = idx[j_idx, :]

        t = (
            np.power(1j, tau - l - 2)
            * sfc[s, n, w]
            / np.sqrt(2 * l * (l + 1))
            * np.exp(
                1j
                * (
                    m * azimuthal_angles[a]
                    - k_medium[w] * np.sum(particles_position[s, :] * e_r[a, :])
                )
            )
        )

        if tau == 1:
            e_1_sca[a, :, w] += t * (
                e_theta[a, :] * pilm[l, np.abs(m), a] * 1j * m
                - e_phi[a, :] * taulm[l, np.abs(m), a]
            )
        else:
            e_1_sca[a, :, w] += t * (
                e_phi[a, :] * pilm[l, np.abs(m), a] * 1j * m
                + e_theta[a, :] * taulm[l, np.abs(m), a]
            )

    return e_1_sca

compute_electric_field_angle_components

Calculates the electric field components in the theta and phi directions for given input parameters.

Parameters:

Name Type Description Default
lmax int

The maximum value of the angular momentum quantum number l. It determines the maximum value of l for which the calculations will be performed.

 required
particles_position ndarray

The positions of particles. It has shape (num_particles, 3), where num_particles is the number of particles and each particle has 3 coordinates (x, y, z).

 required
idx ndarray

A numpy array of shape (jmax, 5), where jmax is the total number of particles multiplied by 2 * lmax * (lmax + 2). Each row of idx represents the indices (s, n, tau, l, m).

 required
sfc ndarray

A 3-dimensional numpy array representing the scattering form factors. It has dimensions (s, n, w).

 required
k_medium ndarray

The wave vector in the medium. It is a numpy array that contains the wave vector values for different frequencies or wavelengths.

 required
azimuthal_angles ndarray

An array representing the azimuthal angles at which the electric field components are computed. It specifies the angles at which the electric field is measured in the azimuthal direction.

 required
e_r ndarray

The unit vector pointing in the direction of the electric field. It is a numpy array of shape (azimuthal_angles.size, 3), where each row corresponds to a different azimuthal angle and the three columns represent the x, y, and z components.

 required
pilm ndarray

A 3-dimensional numpy array of shape (lmax+1, lmax+1, azimuthal_angles.size). It represents the matrix elements of the electric field expansion coefficients for the theta component. The indices (l, m, a) correspond to the spherical harmonics.

 required
taulm ndarray

A numpy array that represents the angular momentum coupling coefficients. It has a shape of (lmax+1, lmax+1, azimuthal_angles.size). The first dimension represents the value of l, the second dimension represents the value of m, and the third dimension represents the azimuthal angle.

 required

Returns:

Name Type Description
e_field_theta ndarray

The electric field component in the theta direction.
e_field_phi ndarray

The electric field component in the phi direction.
Source code in yasfpy/functions/cpu_numba.py 
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@jit(nopython=True, parallel=True, nogil=True, fastmath=True)
def compute_electric_field_angle_components(
    lmax: int,
    particles_position: np.ndarray,
    idx: np.ndarray,
    sfc: np.ndarray,
    k_medium: np.ndarray,
    azimuthal_angles: np.ndarray,
    e_r: np.ndarray,
    pilm: np.ndarray,
    taulm: np.ndarray,
):
    """Calculates the electric field components in the theta and phi directions for given input parameters.

    Args:
        lmax (int): The maximum value of the angular momentum quantum number `l`. It determines the maximum value of `l` for which the calculations will be performed.
        particles_position (np.ndarray): The positions of particles. It has shape `(num_particles, 3)`, where `num_particles` is the number of particles and each particle has 3 coordinates (x, y, z).
        idx (np.ndarray): A numpy array of shape `(jmax, 5)`, where `jmax` is the total number of particles multiplied by `2 * lmax * (lmax + 2)`. Each row of `idx` represents the indices `(s, n, tau, l, m)`.
        sfc (np.ndarray): A 3-dimensional numpy array representing the scattering form factors. It has dimensions `(s, n, w)`.
        k_medium (np.ndarray): The wave vector in the medium. It is a numpy array that contains the wave vector values for different frequencies or wavelengths.
        azimuthal_angles (np.ndarray): An array representing the azimuthal angles at which the electric field components are computed. It specifies the angles at which the electric field is measured in the azimuthal direction.
        e_r (np.ndarray): The unit vector pointing in the direction of the electric field. It is a numpy array of shape `(azimuthal_angles.size, 3)`, where each row corresponds to a different azimuthal angle and the three columns represent the x, y, and z components.
        pilm (np.ndarray): A 3-dimensional numpy array of shape `(lmax+1, lmax+1, azimuthal_angles.size)`. It represents the matrix elements of the electric field expansion coefficients for the theta component. The indices `(l, m, a)` correspond to the spherical harmonics.
        taulm (np.ndarray): A numpy array that represents the angular momentum coupling coefficients. It has a shape of `(lmax+1, lmax+1, azimuthal_angles.size)`. The first dimension represents the value of `l`, the second dimension represents the value of `m`, and the third dimension represents the azimuthal angle.

    Returns:
        e_field_theta (np.ndarray): The electric field component in the theta direction.
        e_field_phi (np.ndarray): The electric field component in the phi direction.
    """
    e_field_theta = np.zeros(
        azimuthal_angles.size * k_medium.size, dtype=complex128
    ).reshape((azimuthal_angles.size, k_medium.size))
    e_field_phi = np.zeros_like(e_field_theta)

    jmax = particles_position.shape[0] * 2 * lmax * (lmax + 2)

    for global_idx in prange(jmax * azimuthal_angles.size * k_medium.size):
        w_idx = global_idx % (jmax * k_medium.size)
        g_idx = global_idx // (jmax * k_medium.size)

        a = g_idx

        w = w_idx % k_medium.size
        j_idx = w_idx // k_medium.size
        s, n, tau, l, m = idx[j_idx, :]

        t = (
            np.power(1j, tau - l - 2)
            * sfc[s, n, w]
            / np.sqrt(2 * l * (l + 1))
            * np.exp(
                1j
                * (
                    m * azimuthal_angles[a]
                    - k_medium[w] * np.sum(particles_position[s, :] * e_r[a, :])
                )
            )
        )

        if tau == 1:
            e_field_theta[a, w] += t * pilm[l, np.abs(m), a] * 1j * m
            e_field_phi[a, w] -= t * taulm[l, np.abs(m), a]
        else:
            e_field_theta[a, w] += t * taulm[l, np.abs(m), a]
            e_field_phi[a, w] += t * pilm[l, np.abs(m), a] * 1j * m

    return e_field_theta, e_field_phi

compute_polarization_components

Compute the polarization components of electromagnetic fields.

Parameters:

Name Type Description Default
number_of_wavelengths int

The number of wavelengths.

 required
number_of_angles int

The number of angles.

 required
e_field_theta ndarray

The electric field component in the theta direction.

 required
e_field_phi ndarray

The electric field component in the phi direction.

 required

Returns:

Name Type Description
degree_of_polarization_tuple tuple

A tuple containing the following polarization components: - I (np.ndarray): The total intensity. - degree_of_polarization (np.ndarray): The degree of polarization. - degree_of_linear_polarization (np.ndarray): The degree of linear polarization. - degree_of_linear_polarization_q (np.ndarray): The degree of linear polarization in the Q direction. - degree_of_linear_polarization_u (np.ndarray): The degree of linear polarization in the U direction. - degree_of_circular_polarization (np.ndarray): The degree of circular polarization.
Source code in yasfpy/functions/cpu_numba.py 
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@jit(nopython=True, parallel=True, nogil=True, fastmath=True)
def compute_polarization_components(
    number_of_wavelengths: int,
    number_of_angles: int,
    e_field_theta: np.ndarray,
    e_field_phi: np.ndarray,
):
    """
    Compute the polarization components of electromagnetic fields.

    Args:
        number_of_wavelengths (int): The number of wavelengths.
        number_of_angles (int): The number of angles.
        e_field_theta (np.ndarray): The electric field component in the theta direction.
        e_field_phi (np.ndarray): The electric field component in the phi direction.

    Returns:
        degree_of_polarization_tuple (tuple): A tuple containing the following polarization components:
            - I (np.ndarray): The total intensity.
            - degree_of_polarization (np.ndarray): The degree of polarization.
            - degree_of_linear_polarization (np.ndarray): The degree of linear polarization.
            - degree_of_linear_polarization_q (np.ndarray): The degree of linear polarization in the Q direction.
            - degree_of_linear_polarization_u (np.ndarray): The degree of linear polarization in the U direction.
            - degree_of_circular_polarization (np.ndarray): The degree of circular polarization.
    """
    # Stokes components
    # S = np.zeros(4 * number_of_angles * number_of_wavelengths, dtype=complex128).reshape((4, number_of_angles, number_of_wavelengths))
    I = np.zeros(number_of_angles * number_of_wavelengths, dtype=float64).reshape(
        (number_of_angles, number_of_wavelengths)
    )
    Q = np.zeros_like(I)
    U = np.zeros_like(I)
    V = np.zeros_like(I)

    for global_idx in prange(number_of_angles * number_of_wavelengths):
        w_idx = global_idx % number_of_wavelengths
        a_idx = global_idx // number_of_wavelengths

        e_field_theta_abs = (
            e_field_theta[a_idx, w_idx].real ** 2
            + e_field_theta[a_idx, w_idx].imag ** 2
        )
        e_field_phi_abs = (
            e_field_phi[a_idx, w_idx].real ** 2 + e_field_phi[a_idx, w_idx].imag ** 2
        )
        e_field_angle_interaction = (
            e_field_theta[a_idx, w_idx] * e_field_phi[a_idx, w_idx].conjugate()
        )

        I[a_idx, w_idx] = e_field_theta_abs + e_field_phi_abs
        Q[a_idx, w_idx] = e_field_theta_abs - e_field_phi_abs
        U[a_idx, w_idx] = -2 * e_field_angle_interaction.real
        V[a_idx, w_idx] = 2 * e_field_angle_interaction.imag

    degree_of_polarization = np.sqrt(Q**2 + U**2 + V**2) / I
    degree_of_linear_polarization = np.sqrt(Q**2 + U**2) / I
    degree_of_linear_polarization_q = -Q / I
    degree_of_linear_polarization_u = U / I
    degree_of_circular_polarization = V / I

    return (
        I,
        degree_of_polarization,
        degree_of_linear_polarization,
        degree_of_linear_polarization_q,
        degree_of_linear_polarization_u,
        degree_of_circular_polarization,
    )

compute_radial_independent_scattered_field

Compute the radial independent scattered field.

Parameters:

Name Type Description Default
number_of_wavelengths int

The number of wavelengths.

 required
number_of_angles int

The number of angles.

 required
e_phi ndarray

The electric field in the phi direction.

 required
e_theta ndarray

The electric field in the theta direction.

 required
e_field_theta ndarray

The electric field theta component.

 required
e_field_phi ndarray

The electric field phi component.

 required

Returns:

Name Type Description
e_1_sca ndarray

The computed radial independent scattered field.
Source code in yasfpy/functions/cpu_numba.py 
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@jit(nopython=True, parallel=True, nogil=True, fastmath=True)
def compute_radial_independent_scattered_field(
    number_of_wavelengths: int,
    number_of_angles: int,
    e_phi: np.ndarray,
    e_theta: np.ndarray,
    e_field_theta: np.ndarray,
    e_field_phi: np.ndarray,
):
    """
    Compute the radial independent scattered field.

    Args:
        number_of_wavelengths (int): The number of wavelengths.
        number_of_angles (int): The number of angles.
        e_phi (np.ndarray): The electric field in the phi direction.
        e_theta (np.ndarray): The electric field in the theta direction.
        e_field_theta (np.ndarray): The electric field theta component.
        e_field_phi (np.ndarray): The electric field phi component.

    Returns:
        e_1_sca (np.ndarray): The computed radial independent scattered field.
    """
    e_1_sca = np.zeros(
        number_of_angles * 3 * number_of_wavelengths, dtype=complex128
    ).reshape((number_of_angles, 3, number_of_wavelengths))

    for global_idx in prange(number_of_angles * number_of_wavelengths):
        w = global_idx % number_of_wavelengths
        a = global_idx // number_of_wavelengths

        e_1_sca[a, :, w] = (
            e_field_theta[a, w] * e_theta[a, :] + e_field_phi[a, w] * e_phi[a, :]
        )

    return e_1_sca

compute_lookup_tables

Compute lookup tables for spherical computations.

Parameters:

Name Type Description Default
lmax int

The maximum degree of the spherical harmonics.

 required
size_parameter ndarray

Array of size parameters.

 required
phi ndarray

Array of azimuthal angles.

 required
cosine_theta ndarray

Array of cosine of polar angles.

 required

Returns:

Name Type Description
spherical_bessel ndarray

Array of spherical Bessel functions.
spherical_hankel ndarray

Array of spherical Hankel functions.
e_j_dm_phi ndarray

Array of exponential terms.
p_lm ndarray

Array of associated Legendre polynomials.
Source code in yasfpy/functions/cpu_numba.py 
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@jit(parallel=True, forceobj=True)
def compute_lookup_tables(
    lmax: int, size_parameter: np.ndarray, phi: np.ndarray, cosine_theta: np.ndarray
):
    """
    Compute lookup tables for spherical computations.

    Args:
        lmax (int): The maximum degree of the spherical harmonics.
        size_parameter (np.ndarray): Array of size parameters.
        phi (np.ndarray): Array of azimuthal angles.
        cosine_theta (np.ndarray): Array of cosine of polar angles.

    Returns:
        spherical_bessel (np.ndarray): Array of spherical Bessel functions.
        spherical_hankel (np.ndarray): Array of spherical Hankel functions.
        e_j_dm_phi (np.ndarray): Array of exponential terms.
        p_lm (np.ndarray): Array of associated Legendre polynomials.
    """
    spherical_hankel = np.zeros(
        (2 * lmax + 1) * np.prod(size_parameter.shape), dtype=complex
    ).reshape((2 * lmax + 1,) + size_parameter.shape)
    spherical_bessel = np.zeros_like(spherical_hankel)
    e_j_dm_phi = np.zeros(
        (4 * lmax + 1) * np.prod(size_parameter.shape[:2]), dtype=complex
    ).reshape((4 * lmax + 1,) + size_parameter.shape[:2])
    p_lm = np.zeros(
        (lmax + 1) * (2 * lmax + 1) * np.prod(size_parameter.shape[:2])
    ).reshape(((lmax + 1) * (2 * lmax + 1),) + size_parameter.shape[:2])

    for p in prange(2 * lmax + 1):
        spherical_hankel[p, :, :, :] = np.sqrt(
            np.divide(
                np.pi / 2,
                size_parameter,
                out=np.zeros_like(size_parameter),
                where=size_parameter != 0,
            )
        ) * hankel1(p + 1 / 2, size_parameter)
        spherical_bessel[p, :, :, :] = spherical_jn(p, size_parameter)
        e_j_dm_phi[p, :, :] = np.exp(1j * (p - 2 * lmax) * phi)
        e_j_dm_phi[p + 2 * lmax, :, :] = np.exp(1j * p * phi)
        for absdm in range(p + 1):
            cml = np.sqrt(
                (2 * p + 1) / 2 / np.prod(np.arange(p - absdm + 1, p + absdm + 1))
            )
            p_lm[p * (p + 1) // 2 + absdm, :, :] = (
                cml * np.power(-1.0, absdm) * lpmv(absdm, p, cosine_theta)
            )

    return spherical_bessel, spherical_hankel, e_j_dm_phi, p_lm

compute_field

Compute the field using the given parameters and coefficients.

Parameters:

Name Type Description Default
lmax int

The maximum degree of the spherical harmonics.

 required
idx ndarray

The index array containing the values of s, n, tau, l, and m.

 required
size_parameter ndarray

The size parameter array.

 required
sph_h ndarray

The spherical harmonics array.

 required
derivative ndarray

The derivative array.

 required
e_j_dm_phi ndarray

The e_j_dm_phi array.

 required
p_lm ndarray

The p_lm array.

 required
pi_lm ndarray

The pi_lm array.

 required
tau_lm ndarray

The tau_lm array.

 required
e_r ndarray

The e_r array.

 required
e_theta ndarray

The e_theta array.

 required
e_phi ndarray

The e_phi array.

 required
scattered_field_coefficients ndarray

The scattered field coefficients array. Defaults to None.

 None
initial_field_coefficients ndarray

The initial field coefficients array. Defaults to None.

 None
scatter_to_internal ndarray

The scatter to internal array. Defaults to None.

 None

Returns:

Name Type Description
field ndarray

The computed field array.
Source code in yasfpy/functions/cpu_numba.py 
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@jit(nopython=True, parallel=True, nogil=True, fastmath=True, cache=True)
def compute_field(
    lmax: int,
    idx: np.ndarray,
    size_parameter: np.ndarray,
    sph_h: np.ndarray,
    derivative: np.ndarray,
    e_j_dm_phi: np.ndarray,
    p_lm: np.ndarray,
    pi_lm: np.ndarray,
    tau_lm: np.ndarray,
    e_r: np.ndarray,
    e_theta: np.ndarray,
    e_phi: np.ndarray,
    scattered_field_coefficients: np.ndarray = None,
    initial_field_coefficients: np.ndarray = None,
    scatter_to_internal: np.ndarray = None,
):
    """
    Compute the field using the given parameters and coefficients.

    Parameters:
        lmax (int): The maximum degree of the spherical harmonics.
        idx (np.ndarray): The index array containing the values of s, n, tau, l, and m.
        size_parameter (np.ndarray): The size parameter array.
        sph_h (np.ndarray): The spherical harmonics array.
        derivative (np.ndarray): The derivative array.
        e_j_dm_phi (np.ndarray): The e_j_dm_phi array.
        p_lm (np.ndarray): The p_lm array.
        pi_lm (np.ndarray): The pi_lm array.
        tau_lm (np.ndarray): The tau_lm array.
        e_r (np.ndarray): The e_r array.
        e_theta (np.ndarray): The e_theta array.
        e_phi (np.ndarray): The e_phi array.
        scattered_field_coefficients (np.ndarray, optional): The scattered field coefficients array. Defaults to None.
        initial_field_coefficients (np.ndarray, optional): The initial field coefficients array. Defaults to None.
        scatter_to_internal (np.ndarray, optional): The scatter to internal array. Defaults to None.

    Returns:
        field (np.ndarray): The computed field array.
    """
    jmax = sph_h.shape[1] * 2 * lmax * (lmax + 2)
    channels = sph_h.shape[-1]

    field = np.zeros(channels * sph_h.shape[2] * 3, dtype=complex128).reshape(
        (channels, sph_h.shape[2], 3)
    )

    if (scattered_field_coefficients is None) and (initial_field_coefficients is None):
        print(
            "At least one, scattered field or initial field coefficients, need to be given."
        )
        print("Returning a zero array")
        return field

    for w_idx in prange(2 * lmax * (lmax + 2) * np.prod(np.array(sph_h.shape[1:]))):
        w = w_idx % channels
        j_idx = w_idx // channels
        sampling_idx = j_idx // jmax
        j_idx = j_idx % jmax
        s, n, tau, l, m = idx[j_idx, :]

        invariant = (
            1 / np.sqrt(2 * (l + 1) * l) * e_j_dm_phi[m + 2 * lmax, s, sampling_idx]
        )
        # Calculate M
        if tau == 1:
            c_term_1 = (
                1j
                * m
                * pi_lm[l, np.abs(m), s, sampling_idx]
                * e_theta[s, sampling_idx, :]
            )
            c_term_2 = tau_lm[l, np.abs(m), s, sampling_idx] * e_phi[s, sampling_idx, :]
            c_term = sph_h[l, s, sampling_idx, w] * (c_term_1 - c_term_2)

            field[w, sampling_idx, :] += (
                scattered_field_coefficients[s, n, w] * invariant * c_term
            )
        # Calculate N
        else:
            p_term = (
                l
                * (l + 1)
                / size_parameter[s, sampling_idx, w]
                * sph_h[l, s, sampling_idx, w]
                * p_lm[l, np.abs(m), s, sampling_idx]
                * e_r[s, sampling_idx, :]
            )
            #   p_term = l * (l + 1) / size_parameter[s, sampling_idx, w]
            #   p_term *= sph_h[l, s, sampling_idx, w]
            #   p_term *= p_lm[l, np.abs(m), s, sampling_idx]
            #   p_term *= e_r[s, sampling_idx, :]

            b_term_1 = (
                derivative[l, s, sampling_idx, w] / size_parameter[s, sampling_idx, w]
            )
            b_term_2 = (
                tau_lm[l, np.abs(m), s, sampling_idx] * e_theta[s, sampling_idx, :]
            )
            b_term_3 = (
                1j
                * m
                * pi_lm[l, np.abs(m), s, sampling_idx]
                * e_phi[s, sampling_idx, :]
            )
            b_term = b_term_1 * (b_term_2 + b_term_3)

            field[w, sampling_idx, :] += (
                scattered_field_coefficients[s, n, w] * invariant * (p_term + b_term)
            )

    return field

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