SDense Format

[jan-david-fischbach]

Hey @flaport,
there seems to be a small oddity in the way dense S-matrices are represented:
The transmission from port 1 to port 2 seems to be S12. This seems to be inconsistent with the matrix expression

$$\mathbf{E}_\mathrm{out} = \mathbf{S} \mathbf{E}_\mathrm{in}$$

where $\mathbf{E}_\mathrm{in, out}$ are in the basis of the port modes. Writing this for a simple single mode two port:

$$\begin{bmatrix} \mathrm{out}_1\\\mathrm{out}_2 \end{bmatrix} = \begin{bmatrix} S_{11}& S_{12} \\\ S_{21}& S_{22} \end{bmatrix} \begin{bmatrix} \mathrm{in}_1\\\mathrm{in}_2 \end{bmatrix}$$

and setting $\mathrm{in}_2=0$. We find $\mathrm{out}2=S{21}\mathrm{in}1$. This is why typically the transmission from 1->2 is contained in $S{21}$ (this is also how it is in meow).

Demo

Let's set up an SDict that contains the S matrix indices compatible to the matrix equation above.

tester = {
        ("1", "1"): 11,
        ("1", "2"): 21,
        ("2", "1"): 12,
        ("2", "2"): 22,
}

The entry with the key ("1", "2") represents the transmission from 1->2 according to the sax tutorials.

S, pm = sax.sdense(tester)
jnp.abs(S)

Array([[11., 21.],
       [12., 22.]], dtype=float64)

which is not the correct matrix indexing...

As far as I can see SDense is not used much within sax. So only minor modifications would be needed to change the convention:

The line converting from SCoo to SDense would need to be adjusted:

sax/sax/saxtypes.py

Line 405 in aabd9f9

S = S.at[..., Si, Sj].add(Sx)

to

S = S.at[..., Sj, Si].add(Sx)

Its inverse would also need to be touched:

sax/sax/saxtypes.py

Line 350 in aabd9f9

Sj, Si = jnp.meshgrid(jnp.arange(S.shape[-1]), jnp.arange(S.shape[-2]))

and in the klu backend:

sax/sax/backends/klu.py

Line 135 in aabd9f9

CextT_S_inv_I_CS_Cext = S_inv_I_CS_Cext[..., Cexti, :][..., :, Cextj]

would need to be adjusted

Conclusions/Questions

It is not clear to me yet, whether this discrepancy between conventions between meow and sax has caused any hidden issues. I'll look into that.

Is there a way we can adjust the convention without introducing loads of regressions? Does it make sense to try, or do we leave it as is?

Best JD

[jan-david-fischbach]

I forgot to mention _sdense_to_sdict, which also needs adjustments

[flaport]

Hi @jan-david-fischbach ,

I think most people work with reciprocal components so I don't expect this change to impact many people (except us as meow developers probably).

Feel free to make a PR if you have the time for it :)