Indexing conventions

The convention is to use

• m: multipole order, -l:l
• l: multipole degree 1:lmax
• s: multipole polarisation type, 1 or 2

so for a T-matrix with \(lmax=3\), we have

l: 1..3
m: -3..3
s: 1..2
900 elements

for each block we introduce a combined p-index such that,

\[ p(n,m) = n(n+1)+m \] which here varies as p: 1..15, as each block is of dimension pmax = lmax(lmax+1)+lmax = 3(3+1)+3 = 15.

The whole T-matrix is indexed with a combined q-index such that,

\[ q(s,p) = (s-1)p_{max} + p \] which here gives us, q: 1..30. The total T-matrix dimension (along rows/columns) is thus \(q_{max}=2\times(lmax(lmax+1)+lmax)=30\).

In summary, for a given \(l_\text{max}\) the indices are given by:

• \(l=1:l_\text{max}\)
• \(m=-l:l\) for each l
• \(s=1:2\) for each pair of (l,m)
• \(p(l,m) = l(l+1)+m\)
• \(q(s,p) = (s-1)p_{max} + p\)

Given \(q\) and \(l_\text{max}\) we can invert these indices as follows,

• \(p = q - (s-1)q_\text{max}/2\),
  giving here, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15
• \(l = \lfloor\sqrt{p}\rfloor\),
  giving here 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3
• \(m = p - l(l+1)\),
  giving here -1, 0, 1, -2, -1, 0, 1, 2, -3, -2, -1, 0, 1, 2, 3, -1, 0, 1, -2, -1, 0, 1, 2, -3, -2, -1, 0, 1, 2, 3

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30
p	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
l	1	1	1	2	2	2	2	2	3	3	3	3	3	3	3	1	1	1	2	2	2	2	2	3	3	3	3	3	3	3
m	-1	0	1	-2	-1	0	1	2	-3	-2	-1	0	1	2	3	-1	0	1	-2	-1	0	1	2	-3	-2	-1	0	1	2	3
s	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2