Improving the interface (substrate) implementation

[johnaparker]

Currently, MiePy utilizes the normal incidence approximation (NIA) for dealing with a substrate. The exact solution is outlined in [1].

Implementation

• Write the current NIA approximation in C++ (current implementation is slow)
• With miepy.interface, add a bool parameter nia = True to toggle NIA (default True until exact solution is implemented)
• Implement the exact solution in C++

Tests

• Verify the NIA approximation with the exact solution for a single particle or multiple particles far from the interface
• Compare to FDTD simulations with a substrate

Notes

• Calculating the E and H field can be done in the NIA approximation using field expansions around the mirror particles. It is less clear how to do so in the exact solution (sec. 2.2.2 in [1] discusses this)
• The definition of the cross-sections in the presence of the interface is unclear (sec. 2.5 in [1])

References

1. Mackowski_2008_Exact solution for the scattering and absorption properties of sphere clusters.pdf
   Eqs.(19-33) for exact solution, Eq.(34) for NIA