Mode solver sweeps using interpolation

[dbochkov-flexcompute]

Is your feature request related to a problem? Please describe.
Sometimes it is of interest to obtain the waveguide mode field profiles and/or propagation constants at a large number of frequency points. Given that waveguide modes often exhibit smooth frequency dependence, it could be much faster and still accurate to solve for waveguide modes only at a smaller number of selected frequencies and then interpolate found solutions onto frequency points in-between, as opposed to directly solving mode equations at each frequency point.

Describe the solution you'd like
Various types of interpolations such as piecewise linear, quadratic, spline, etc can be considered. In the first version of such a feature it probably makes most sense to use Chebyshev interpolation given its high accuracy. One approach to implement this consists in adding a field sweep_spec in the ModeSolver class, that could take values of ModeSweepSpecType = Union[DirectSolveSpec, ChebyshevInterpSpec] (default: DirectSolveSpec()). Internally, ModeSolver should wrap this part of code https://github.com/flexcompute/tidy3d/blob/develop/tidy3d/components/mode/mode_solver.py#L463-L464 such that if sweep_spec is an instance of class ChebyshevInterpSpec:

• it detects the range of frequency sample points where the mode solutions are requested
• create an auxiliary ModeSolver instance that differs from the original one only by value of field freqs, which now would be chebyshev nodes in the detected frequency range (the number of chebyshev nodes to use should be a field in ChebyshevInterpSpec class)
• retrieves ModeSolverData from this auxiliary ModeSolver
• applies mode tracking to ModeSolverData to facilitate correct interpolation
• interpolates all quantities (propagation index, fields, grid correction factors, etc) onto the original frequency sample points and constructs a new ModeSolverData based on these interpolated values.

Additional context

• https://arxiv.org/pdf/1810.04282
• https://docs.simulation.cloud/projects/tidy3d/en/latest/notebooks/ModeSolver.html#Mode-tracking
• https://docs.simulation.cloud/projects/tidy3d/en/latest/notebooks/ModalSourcesMonitors.html#Waveguide-junction-(using-broadband-source)

[doddgray]

I second that this would be useful and would advocate for facilitating interpolation between mode solutions as a function of more general parameters than optical frequency, especially waveguide geometry parameters (width, thickness, cladding thickness, coupled wg gap) and material parameters (index, composition, crystal angle, temperature, etc) which are often varied in practice to fine tune modal properties.

That said I think this is tricky and generally impossible to make 100% robust because mode crossings invalidate the assumption of smooth variation and the well-definedness of which mode is which locally near a-priori-unknown regions of parameter space (including optical frequency). I work on waveguides with lots of mode crossings and have found this type of issue causes interpolations and mode tracking to generally be unreliable. I have found in my cases that labeling/filtering to the mode data based on heuristic analysis of the mode fields (eg quasi TE00, TM00, TE01, ... based on overlap integrals with hermite Gaussians) and flagging data too close to identified mode crossings (modes too mixed, group index spikes) so that interpolations are not performed across these regions of parameter space can help a lot.