tutorial example on breakdown of glancing-angle mode launcher?

[stevengj]

Although we support eigenmode sources (and mode coefficients) for oblique waveguides (see #698 and #940), of course problems will arise as the waveguide approaches glancing incidence to the source plane:

1. The intersection of the waveguide with the source plane becomes larger and larger, necessitating a bigger computational cell.

2. Modal dispersion becomes a problem: the sensitivity of the field pattern (in the source plane) to frequency becomes larger and larger, so an eigenmode source will become narrower bandwidth. Probably discretization errors become worse as well.

It might be nice to have a tutorial example illustrating this, to give a rough rule of thumb of the maximum angle where this is practical.

(Of course, for angles greater than 45° you will always be better off rotating the source plane, but in some circumstances this might be inconvenient, e.g. if you are continuously optimizing the waveguide angle.)

[oskooi]

As shown in the field profiles below for waveguide rotation angles of 0°, 20°, 40°, 60°, 80°, and 100°, there does not seem to be any noticeable distortion in the mode as the waveguide approaches glancing incidence to the source plane. (Note that the field profiles of the single-mode waveguide were obtained using a CW source.) Additionally, the total power in the waveguide mode (as measured by a flux monitor using a pulse source) varies less than 5% at 80° relative to 0°.

As long as the line source/monitor fully intersects the waveguide, it seems that the mode can be properly launched/decomposed. In this example, the maximum waveguide rotation is ~83°.

The script used to generate these results is based on Tutorials/Eigenmode Source/Index-Guided Modes of a Ridge Waveguide:

import meep as mp
import numpy as np
import matplotlib
matplotlib.use('agg')
import matplotlib.pyplot as plt

resolution = 50 # pixels/μm                                                                                                                                                                   

cell_size = mp.Vector3(14,14)

pml_layers = [mp.PML(thickness=2)]

# rotation angle (in degrees) of waveguide, counter clockwise (CCW) around z-axis                                                                                                             
rot_angle = np.radians(80)

w = 1.0 # width of waveguide                                                                                                                                                                  

geometry = [mp.Block(center=mp.Vector3(),
                     size=mp.Vector3(mp.inf,w,mp.inf),
                     e1=mp.Vector3(1).rotate(mp.Vector3(z=1), rot_angle),
                     e2=mp.Vector3(y=1).rotate(mp.Vector3(z=1), rot_angle),
                     material=mp.Medium(epsilon=12))]

fsrc = 0.15 # frequency of eigenmode or constant-amplitude source                                                                                                                             
kx = 0.4    # initial guess for wavevector in x-direction of eigenmode                                                                                                                        
bnum = 1    # band number of eigenmode                                                                                                                                                        

kpoint = mp.Vector3(kx).rotate(mp.Vector3(z=1), rot_angle)

compute_flux = False # compute flux (True) or plot the field profile (False)                                                                                                                   

sources = [mp.EigenModeSource(src=mp.GaussianSource(fsrc,fwidth=0.2*fsrc) if compute_flux else mp.ContinuousSource(fsrc),
                              center=mp.Vector3(0),
                              size=mp.Vector3(y=14),
                              direction=mp.NO_DIRECTION,
                              eig_kpoint=kpoint,
                              eig_band=bnum,
                              eig_match_freq=True)]

sim = mp.Simulation(cell_size=cell_size,
                    resolution=resolution,
                    boundary_layers=pml_layers,
                    sources=sources,
                    geometry=geometry)

if compute_flux:
    tran = sim.add_flux(fsrc, 0, 1, mp.FluxRegion(center=mp.Vector3(x=5), size=mp.Vector3(y=10)) if rot_angle < np.pi/4 else mp.FluxRegion(center=mp.Vector3(y=5), size=mp.Vector3(10)))
    sim.run(until_after_sources=50)
    res = sim.get_eigenmode_coefficients(tran,
                                         [1],
                                         direction=mp.NO_DIRECTION,
                                         kpoint_func=lambda f,n: kpoint)
    print("flux:, {:.6f}, {:.6f}".format(mp.get_fluxes(tran)[0],abs(res.alpha[0,0,0])**2))
else:
    sim.run(until=100)
    sim.plot2D(output_plane=mp.Volume(center=mp.Vector3(), size=mp.Vector3(10,10)),
               fields=mp.Ez,
               field_parameters={'alpha':0.9})
    plt.title("rotation angle = {}°".format(int(np.ceil(np.degrees(rot_angle)))))
    plt.savefig("oblique_rotangle{}.png".format(int(np.ceil(np.degrees(rot_angle)))))

[stevengj]

I'm guessing that if you look at the flux at several frequencies, I'm guessing that it is more distorted at other frequencies as you go towards glancing incidence.

But it is nice to see that it is working so well! I was worried that you would start to see discretization effects by these angles.

[stevengj]

(At some point you will see discretization problems even if you have a large enough cell.)

[stevengj]

Note that only the eig_kpoint direction affects MPB (it changes one of the lattice vectors). MPB does not look at the block orientation directly (it only cares about the materials in the source plane, regardless of how they are created).

[oskooi]

There does not seem to be much difference in the relative error in the Poynting flux vs. rotation angle of the waveguide at different frequencies using a broadband source. The error (relative to 0°) increases with rotation angle as expected but the increase is independent of the frequency.

These results were generated using a y-directed line monitor which is why the maximum angle is 45° (above 45° and the waveguide no longer fully intersects the vertical line monitor). Note also the semilog scale.