The Hansen Multipoles

We have already seen how if we let E or B be given by

$${\bf E} \mbox{\rm  or  } {\bf B} = \frac{1}{\sqrt{\ell(\ell + 1)}} f_\ell(kr) {\bf L} Y_L(\hat{r})$$ (13.1)

then

1. Both the fields given above and their partner fields (given by the curl) have zero divergence.
2. The fields given above are completely transverse, since $\hat{r} \cdot {\bf L} = 0$ (operator).
3. The partner fields given by the curl are not purely transverse.
4. In order to be consistent, the fields above are also the curls of the partner fields. In fact, this follows from vector identities for divergenceless fields.

It is therefore sensible to define, once and for all, a set of multipoles that embody these properties. In addition, anticipating a need to treat longitudinal fields as well as transverse fields, we will define a third kind of multipoles with zero curl but non-zero divergence. These will necessarily be ``connected'' to sources (why?). We will call these ``pre-computed'' combinations of bessel functions, vector spherical harmonics, and their curls the Hansen Multipoles (following unpublished notes from L. C. Biedenharn as I have been unable to determine his original reference):