[7/13][Adjoint Module] Launch a mode through a stratified stack into the simulation (and backprop)

[smartalecH]

Launches a mode into a simulation through a stratified stack, completing the angular-spectrum work in #3281 (which carries fields out through a stack). The typical use is a fiber far above a chip: the mode is specified where it physically lives, at the fiber facet outside the FDTD cell, carried analytically through the superstrate, and injected just above the device. The whole path is differentiable, so the fiber's waist, tilt and offset can be optimized together with the device.

API.

propagator = mpa.AngularSpectrum.from_volume(sim, plane, stack, [fcen])
fields = propagator.incident_fields(mpa.gaussian_mode(waist, tilt_deg=tilt), distance=d)
sim.sources += propagator.equivalent_sources(fields, mp.GaussianSource(fcen, fwidth=df),
                                             center=plane.center, size=plane.size)

• incident_fields(mode, distance) gives the tangential fields at the injection plane for a mode arriving from the far side of the stack.
  • It is the time reverse of the measurement path, so the same Mode describes the same fiber in both directions: what overlap collects with gaussian_mode(tilt=θ) is what incident_fields launches with it. Its spectrum is conj(a(-kt)).
  • The stack is crossed from the far side, with the reciprocal transmission t · Y_far / Y_monitor.
• equivalent_sources(fields, ...) turns those fields into an electric sheet K = n̂×H and a magnetic sheet N = −n̂×E, which radiate one way, as ordinary amp_data sources.
• equivalent_currents(fields) returns the same arrays in JAX. A loss can feed them to MeepJaxWrapper and differentiate end to end with respect to the mode's parameters; Meep returns the cotangent with respect to amp_data through [6/13][Adjoint Module] Adjoint gradients with respect to sources #3283's source adjoint.
• from_volume builds a propagator for a plane that has no monitor yet, since the sources are exactly what it is building.

End-to-end check (examples/adjoint_optimization/grating_coupler_asm_end_to_end.py): a small 2D grating coupler with the fiber 10 µm up in air, above 2 µm of silica, all outside the cell.

• (a, b) Field: Meep's launched field matches the propagator's prediction 3.5 µm below the injection plane to 2.5%, with no fitted scale; the only factor applied is the source's own spectrum.
• (c) Gradient: the gradient of the coupled waveguide power with respect to the fiber's waist, tilt and offset matches finite differences of the full simulation to ≤ 1e-4.
• (d) Reciprocity: fiber → waveguide and waveguide → fiber efficiencies agree across fiber tilt.
• (e) Convergence: the remaining gap is discretization error, shrinking at second order with resolution (28% → 16% → 8.5% at 20/30/40 px/µm).

The example runs in about 20 minutes on one core, or about 2 minutes with --quick.

Tests.

• Injection energy through an interface: the launched power equals the reciprocal power transmission times the mode power.
• A tilted mode is launched in the direction the fiber emits.
• The time-reversed injected beam projects onto the mode with purity 1.
• equivalent_currents equals the placed amp_data and is differentiable with respect to the mode.
• x- and y-normal planes in 2D, and the amp_data layout for every normal in 3D.
• One-way radiation in FDTD.

Also includes the example grating_coupler_asm_source.py, which drives the coupler from the fiber and co-optimizes the grating and the fiber alignment.