Spherical coordinates

links

• Legendre poly
• scipy spherical harmonics
• spherical coordinates
• $\grad\times$ in spherical
• Spherical Bessel and Hankel functions

Plane wave with spherical harmonics expansion

Scalar normal incidence

A scalar plane wave along $z$ is expanded upon spherical waves using : $$\begin{equation} E_x = e^{jkz} =\sum_{n=0}^{\infty} a_nJ_n(kr)P_n(\cos\theta) \end{equation}$$ where $J_n$ are the spherical Bessel functions of the first kind, $P_n$ the associated Legendre polynomials of the first kind and $a_n=j^{-n}(2n+1)$.

A plane wave expansion upon spherical waves, with $\lambda=1$ over y,z=[-5,5].

N=10	N=20	N=50	N=99

Scalar at oblique incidence

A scalar plane wave at an angle $\alpha=20^{\circ}$ is expanded upon spherical harmonics using wiki: $$\begin{equation} E_x(r,\theta,\phi) = 4\pi \sum_{n=0}^{\infty}\sum_{m=-n}^{n} j^n J_n(kr)Y_n^{m*}(\theta,\phi)Y_n^m(\alpha,0) \end{equation}$$

N=10	N=30	N=40	N=50

Translational addition theorem

• python sage wigner3j symbols
• translational addition theorem

Scalar

Translates the spherical wave function $\psi_{l,m}$ from reference $O(\bb r)$ to $O^{'}$ where $\bb t=\bb r^{'} - \bb r$. $$\begin{eqnarray} \psi_{l,m}^{(out)}(\bb r^{'}) &=& \sum_{\nu=0}^{\infty}\sum_{\mu=-n}^{n} a_{l,m;\nu,\mu}(\bb t)^{(out-in)}\psi_{\nu,\mu}^{(in)}(\bb r)\\ a_{l,m;\nu,\mu}^{out-in}(\bb t) &=& 4\pi\sum_{q=|l-\nu|}^{l+\nu} \left(-i\right)^{l-\nu-q}\psi^{(out)}_{q,m-\mu}(\bb t) \left(-1\right)^m\cc G(l,\nu,q,m,-\mu,-m+\mu) \end{eqnarray}$$ where $\psi_{l,m} = z_l(r)Y_l^m(\theta,\phi)$ and $z_l=j_l$ for $\psi^{(in)}$ and $z_l=h_l^{(1)}$ for $\psi^{(out)}$ and
$\cc G$ are the Gaunt coefficients.

Application for a few $\psi_{l,m}$ the case $\bb t=d\bb{e_r}= d\left(\cos\theta\bb{e_z} + \sin\theta\bb{e_y}\right)$

Normal Hankel functions of the first kind.

$\psi_{00}$	$\psi_{10}$	$\psi_{20}$	$\psi_{30}$

Translated Bessel functions of the first kind.

$\psi_{00}$	$\psi_{10}$	$\psi_{11}$	$\psi_{20}$