Fix full modal nonlinear polarisation

[chrisbrahms]

This should (I think) fix #246

dimlimits for MarcatiliModes are now (0, π) rather than (0, 2π). There is a new function Modes.geomfac which takes this into account by multiplying the integral in TransModal by 2. For other modes this defaults to 1. With this change, the initial points at which pcubature evaluates the integrand are now (0, π/2, π), so it no longer gets stuck.

I still have to fully convince myself that this is general--so far it produces correct results for both LP11 propagation (new test in test_polarisation_field.jl and HE11 propagation (test_multimode.jl).

As it stands this will mess up derived modes, like for example a ZeisbergerMode, because they don't implement geomfac. So the brute-force method of calculating Modes.N and Modes.Aeff no longer produces correct results.

TODO:

• More comprehensive tests
• Move multiplication by geomfac into the loop in TransModal to avoid the extra array multiplication
• Think about a better way of working out the multiplication factor to keep dimlimits the same

[chrisbrahms]

Ok this definitely does not work. I get the wrong answer when pumping in LP11 but with HE11 present. With Kerr only, there should be no nonlinear polarisation in HE11, but I get the same as in HE21 and TM01:

(the modes here are TM01, HE21, HE11)

This is because HE11 and HE21/TM01 don't have the same symmetry, so we do actually need to integrate (0, 2π) for the integral to vanish.

I think our best option for now is actually to switch to hcubature. A quick test with just two modes (HE21 and TM01 as I've been using) shows it's actually faster. With this PR:

julia> @btime transform_vert(nl_vert, Eω_vert, 0.0);
  266.604 ms (8471 allocations: 99.19 MiB)
julia> transform_vert.ncalls
527

Without this PR, but switching to hcubature_v in TransModal for full 2D integrals:

julia> @btime transform_vert(nl_vert, Eω_vert, 0.0);
  155.324 ms (5195 allocations: 60.84 MiB)
julia> transform_vert.ncalls
323

I think it's possible that when @jtravs measured the performance of pcubature_v vs hcubature_v before, pcubature_v was coming out faster simply because it was ignoring most of the integration domain.

[jtravs]

I'm much happier with this, and I think you're correct about my benchmarks. I will review once I see the tests have passed.

[jtravs]

Wait, why are tests skipped?

[chrisbrahms]

Tests were skipped because I marked this as draft. They're running now.

[chrisbrahms]

There is still a stack overflow error on the v1.5 tests, but I'm pretty certain it's not this PR (see #248 ) so I'll merge.

I think the error is due to one of our dependencies getting a minor update. Need to check in more detail.