treams.special.tl_vsw_B

treams.special.tl_vsw_B(lambda, mu, l, m, x, theta, phi) = <ufunc 'tl_vsw_B'>

Translation coefficient for vector spherical waves with opposite parity

Definded by [1]

\[\begin{split}B_{\lambda\mu lm}^{(3)}(x, \theta, \varphi) = \frac{\gamma_{lm}}{\gamma_{\lambda\mu}} (-1)^m \frac{2\lambda + 1}{\lambda (\lambda + 1)} \mathrm i^{\lambda - l} \sqrt{\pi \frac{(l + m)!(\lambda - \mu)!}{(l - m)!(\lambda + \mu)!}} \\ \cdot \sum_{p} \mathrm i^p\sqrt{2p + 1} h_p^{(1)}(kr) Y_{p, m - \mu}(\theta, \varphi) \begin{pmatrix} l & \lambda & p \\ m & -\mu & -m + \mu \end{pmatrix} \\ \cdot \begin{pmatrix} l & \lambda & p - 1 \\ 0 & 0 & 0 \end{pmatrix} \sqrt{\left[(l + \lambda + 1)^2 - p^2\right]\left[p^2 - (l - \lambda)^2\right]}\end{split}\]

with

\[\gamma_{lm} = \mathrm i \sqrt{\frac{2l + 1}{4\pi l (l + 1)}\frac{(l - m)!}{(l + m)!}}\]

and the Winger 3j-symbols (treams.special.wigner3j()) and the spherical Hankel functions. The summation runs over all \(p \in \{\lambda + l - 1, \lambda + l - 3, \dots, \max(|\lambda - l| + 1, |\mu - m|)\}\).

These coefficients are used to translate from scattered to incident modes.

Parameters:

  • lambda (integer, array_like) – Degree of the destination mode

  • mu (integer, array_like) – Order of the destination mode

  • l (integer, array_like) – Degree of the source mode

  • m (integer, array_like) – Order of the source mode

  • x (complex, array_like) – Translation in units of the wave number

  • theta (float or complex, array_like) – Polar angle

  • phi (float, array_like) – Azimuthal angel

Returns:

complex

References