propagator settings

3. propagator settings#

3.1. \(z\)-stepping parameters for characterization#

Propagator.characterize() applies an adaptive algorithm to select the \(z\) step size between calculations. This algorithm is as follows. At a given step, we interpolate through the eigenmodes at the last few \(z\) values and compare the current eigenmodes with the extrapolated prediction. If the error is below a threshhold controlled by z_acc, then the step is accepted, and if the error is less than a tenth the threshhold, the next step is doubled; otherwise, the step is divided in two and the couping-coefficient matrix is re-computed at this new step. z_acc and other parameters are accessible as Propagator class attributes, listed below:

  • z_acc : this parameter controls the adaptive stepping. Higher = more accurate; the scale is logarithmic, and the default value is 0.

  • fixed_zstep : set this to a numeric value to use a fixed \(z\) step, bypassing the adaptive scheme.

  • min_zstep : this is the minimum \(z\) step that can be chosen by the adaptive scheme, default 1.25.

  • max_zstep : this is the maximum \(z\) step that can be chosen by the adaptive scheme. This is useful to prevent the adaptive stepper from skippin over small, peaked features in the cross-coupling matrix, default 640.

  • init_zstep : the starting \(z\) step; defaults to 10.

3.2. specifying degenerate modes#

The presence of degenerate modes in a waveguide will typically slow down computation and can sometimes cause issues with numerical stability. This is because such modes, as computed by the finite-element solver, can “rotate” rapidly as we advance in \(z\). This produces sharp peaks in the coupling coefficients.

Note

From my testing, a rule of thumb is that modes can be treated as degenerate if the difference in effective index is less than \(10^{-5}\).

If we know that certain modes will remain degenerate throughout the entire waveguide (e.g. we ran Propagator.compute_neffs() before hand), we can pass that information into the following Propagator attribute.

  • degen_groups : a nested list of indices which identify groups of modes that remain degenerate throughout the entire waveguide. For instance, if we know a priori that modes (1,2) and modes (3,4) form degenerate pairs, we could pass in [[1,2],[3,4]]. This is highly recommended whenever possible.

Given this information, the code will apply a change of basis to each degenerate mode group which minimizes rotation with \(z\).

What if the mode degeneracies are not consistent through the waveguide? As far as I can tell, this is pretty tricky to simulate, especially in cases where eigenvalues split and converge repeatedly. Thus, my official advice (for now) is to break up such a waveguide into regions in \(z\) where the degeneracy structure is consistent, and perform the characterizations and propagations in parts. See 19-port photonic lantern for an example.

3.3. accessing the data#

After running characterize(), you may want to take a closer look at how the eigenvalues, eigenmodes, and coupling coefficients change with \(z\). This data is returned by characterize() but is also stored in the following variables:

  • Propagator.zs : the array of \(z\) values traversed by characterize().

  • Propagator.neffs : the effective refractive indices of the eigenmodes. The mode effective index \(n_e\) is related to the eigenvalue \(w\) by \(n_e = \sqrt{w/k^2}\), where \(k\) is the free space wavenumber.

  • Propagator.vs : the eigenmode basis computed over zs.

  • Propagator.cmats : the coupling coefficient matrix computed over zs.

If save=True was set, the above arrays are also saved to file and can be loaded with Propagator.load().

This data may be plotted using the following functions.

  • plot_neffs() : plot the effective indices.

  • plot_waveguide_mode(i) : plot eigenmode i; the plot comes with a slider for \(z\).

  • plot_coupling_coeffs() : plot the coupling coefficients between the modes.

Interpolation functions for the above data are automatically generated, but you can remake these functions manually with

  • Propagator.make_interp_funcs(zs)

which is useful if you want to transform or rescale the \(z\) coordinate. These interpolation functions are at

  • Propagator.compute_neff(z)

  • Propagator.compute_v(z)

  • Propagator.compute_cmat(z)

3.4. propagation parameters#

To solve the coupled-mode equations, cbeam uses scipy.integrate.solve_IVP(). The integration scheme used by this function can be set through the class attribute

  • Propagator.solver : the scheme (the “method” argument) to pass to solve_IVP(); default is "RK45" (4th order adaptive Runge-Kutta).

Whe propagating, there is also the option to include a minor WKB-like correction to the coupled-mode equations. Set this through the attribute

  • Propagator.WKB (bool): this controls whether or not the correction is included in the coupled-mode equations (default False).

The effect of this correction is typically negligible, and becomes important only if mode propagation constants change significantly through the waveguide.