VectorSphericalWaveFunction¶

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Specifies an illuminating, time-harmonic vector spherical wave function,

$\begin{eqnarray*} \VField{\VField{E}} & = & a\pvec{\{N,M\}}^{1}_{nm}(\pvec{r},k_+) \end{eqnarray*}$

To specify a vector spherical wave function illumination the following parameters are required:

the scaling Coefficient $a$ ,
the angular frequency Omega or vacuum wavelength Lambda0,
the integer multipole degree n, order m and type (N or M) of the vector spherical wave function $\pvec{N}_{nm},\pvec{M}_{nm}$ ,

SourceBag {
  Source {
    ElectricFieldStrength {
      VectorSphericalWaveFunction {
        Coefficient = 1
        Lambda0 = 50e-9
        MultipoleDegree = 1
        MultipoleOrder = -1
        Type = M
      }
    }
  }
}

Theoretical background

It is required that the exterior of the computational domain is a lossless, homogeneous and isotropic material distribution enclosing the origin of the vector spherical wave function. Let $\varepsilon_+$ and $\mu_+$ denote the corresponding scalar permittivity and permeability, respectively. The angular wave number is given by $k_+=\omega \sqrt{\varepsilon_+ \mu_+}$ .

The vector spherical wave functions $\pvec{M}_{nm},\pvec{N}_{nm}$ for the incoming fields have the following definition [1] in terms of spherical coordinates $(r,\vartheta,\varphi)$

$\begin{eqnarray*} \pvec{M}^{1}_{nm} = \gamma_{nm} j_{n}(kr)\nabla\times\left(\pvec{r} P_{n}^m(cos(\vartheta))e^{im\varphi}\right) \end{eqnarray*}$

$\begin{eqnarray*} \pvec{N}^{1}_{nm} = \frac{1}{k} \nabla \times \pvec{M}^{1}_{nm} = \gamma_{nm} \left[ \frac{n(n+1)}{kr}j_{n}(kr) \frac{\pvec{r}}{r} P_{n}^m(cos(\vartheta))e^{im\varphi} + \frac{r}{kr} \frac{d}{d(kr)}\left(kr j_n(kr)\right) \nabla \left( P_{n}^m(cos(\vartheta))e^{im\varphi} \right) \right] \end{eqnarray*}$

with the common normalization factor

$\begin{eqnarray*} \gamma_{nm} & = & \sqrt{\frac{(2n+1)(n-m)!}{4\pi n (n+1)(n+m)!}}. \end{eqnarray*}$

The definition makes use of the spherical Bessel functions $j_n(x)$ and the associated Legendre polynomials $P_n^m(x)$ of degree $n$ and order $m$

Bibliography

[1]	Mishchenko, Michael I., Larry D. Travis, and Andrew A. Lacis. Scattering, absorption, and emission of light by small particles. Cambridge university press, 2002.