glmtThe glmt class implements Mie theory within the
Generalized Lorenz-Mie Theory (GLMT) framework to compute scattering
coefficients for a spherical scatterer. It calculates the Mie
coefficients \(a_j\), \(b_j\), \(c_j\), and \(d_j\) for electric and magnetic multipoles,
supporting both one-dimensional and two-dimensional parameter spaces
(e.g., refractive index and size parameter). The class precomputes
spherical Bessel and Hankel functions for efficiency. Expressions for
the Mie coefficients are taken from Ch. 4 in Bohren(Craig F.Bohren 2004).
Creating an instance of the glmt class initializes the
Mie coefficients and precomputes spherical Bessel and Hankel functions
for the specified parameters. The class supports both scalar and array
inputs for flexibility in analyzing multiple configurations.
glmt(maxJ, wl, nr, x, mu=1, mu1=1, dim=None)int — maximum total angular momentum quantum number.
float or array — wavelength of the incident field (in microns).
float or array — relative refractive index of the scatterer.
float or array — size parameter (\(k r\), where \(k = 2\pi / \lambda\) and \(r\) is the scatterer radius).
float — relative permeability of the scatterer (default is 1).
float — relative permeability of the medium (default is 1).
int or None — dimensionality of the parameter space: 1
for coupled nr and wl arrays,
None for 2D meshgrid (default is None).
Computes the Mie scattering coefficient \(a_j\) for electric multipoles up to
maxJ. The coefficient is given by: \[a_j = \frac{\mu n_r^2 j_j(n_r x) [x j_j(x)]'
- \mu_1 j_j(x) [n_r x j_j(n_r x)]'}{\mu n_r^2 j_j(n_r x) [x
h_j(x)]' - \mu_1 h_j(x) [n_r x j_j(n_r x)]'},\] where
\(j_j\) is the spherical Bessel
function, \(h_j\) is the spherical
Hankel function of the first kind, \(n_r\) is the relative refractive index,
\(\mu\) and \(\mu_1\) are permeabilities, and \([f(x)]' = f(x) + x f'(x)\). Returns
an array of shape (maxJ+1, ...).
Computes the Mie scattering coefficient \(b_j\) for magnetic multipoles up to
maxJ. The coefficient is given by: \[b_j = \frac{\mu_1 j_j(n_r x) [x j_j(x)]' -
\mu j_j(x) [n_r x j_j(n_r x)]'}{\mu_1 j_j(n_r x) [x h_j(x)]' -
\mu h_j(x) [n_r x j_j(n_r x)]'},\] where variables are as
defined for \(a_j\). Returns an array
of shape (maxJ+1, ...).
Computes the Mie internal coefficient \(c_j\) for magnetic multipoles up to
maxJ. The coefficient is given by: \[c_j = \frac{\mu_1 j_j(x) [x h_j(x)]' - \mu_1
h_j(x) [x j_j(x)]'}{\mu_1 j_j(n_r x) [x h_j(x)]' - \mu h_j(x)
[n_r x j_j(n_r x)]'},\] where variables are as defined for
\(a_j\). Returns an array of shape
(maxJ+1, ...).
Computes the Mie internal coefficient \(d_j\) for electric multipoles up to
maxJ. The coefficient is given by: \[d_j = \frac{\mu_1 n_r j_j(x) [x h_j(x)]' -
\mu_1 n_r h_j(x) [x j_j(x)]'}{\mu n_r^2 j_j(n_r x) [x h_j(x)]' -
\mu_1 h_j(x) [n_r x j_j(n_r x)]'},\] where variables are as
defined for \(a_j\). Returns an array
of shape (maxJ+1, ...).
Computes the spherical Hankel function of the first kind \(h_j(x) = j_j(x) + i y_j(x)\), or its
derivative if derivative=True, where \(j_j\) and \(y_j\) are spherical Bessel functions of the
first and second kinds, respectively.
The glmt class efficiently precomputes spherical Bessel
and Hankel functions to calculate Mie coefficients for scattering and
internal fields. It supports both 1D (coupled wavelength and refractive
index) and 2D (meshgrid of size parameter and refractive index)
parameter spaces. Ensure that wl, nr, and
x are positive to avoid invalid physical
configurations.