Mode Solver#

Run this notebook in your browser using Binder.

This tutorial shows how to use the mode solver plugin in tidy3d.

[1]:
import numpy as np
import matplotlib.pylab as plt

import tidy3d as td
from tidy3d.constants import C_0
import tidy3d.web as web

from tidy3d.plugins import ModeSolver

Setup#

We first set up the mode solver with information about our system. We start by setting parameters

[2]:
# size of simulation domain
Lx, Ly, Lz = 6, 6, 6
dl = .05

# waveguide information
wg_width = 1.5
wg_height = 1.0
wg_permittivity = 4.0

# central frequency
wvl_um = 2.0
freq0 = C_0 / wvl_um
fwidth = freq0 / 3

# run_time in ps
run_time = 1e-12

# automatic grid specification
grid_spec = td.GridSpec.auto(min_steps_per_wvl=20, wavelength=wvl_um)

Then we set up a simulation, in this case including a straight waveguide and default periodic boundary conditions.

[3]:
waveguide = td.Structure(
    geometry = td.Box(size=(wg_width, td.inf, wg_height)),
    medium = td.Medium(permittivity=wg_permittivity)
)

sim = td.Simulation(
    size=(Lx, Ly, Lz),
    grid_spec=grid_spec,
    structures=[waveguide],
    run_time = run_time,
)

ax = sim.plot(z=0)
[09:13:36] WARNING  No sources in simulation.                                                     simulation.py:420
../_images/notebooks_ModeSolver_5_1.png

Initialize Mode Solver#

With our system defined, we can now create our mode solver. We first need to specify on what plane we want to solve the modes using a td.Box() object.

[4]:
plane = td.Box(
    center=(0,0,0),
    size=(4, 0, 3.5)
)

The mode solver can now compute the modes given a ModeSpec object that specifies everything about the modes we’re looking for, for example:

  • num_modes: how many modes to compute.

  • target_neff: float, default=None, initial guess for the effective index of the mode; if not specified, the modes with the largest real part of the effective index are computed.

The full list of specification parameters can be found here.

[5]:
mode_spec = td.ModeSpec(
    num_modes=3,
    target_neff=2.0,
)

We can also specify a list of frequencies at which to solvefor the modes.

[6]:
num_freqs = 11
f0_ind = num_freqs // 2
freqs = np.linspace(freq0 - fwidth / 2, freq0 + fwidth / 2, num_freqs)

Finally, we can initialize the ModeSolver, and call the solve method.

[7]:
mode_solver = ModeSolver(
    simulation=sim,
    plane=plane,
    mode_spec=mode_spec,
    freqs=freqs,
)
mode_data = mode_solver.solve()

Visualizing Mode Data#

The mode_info object contains information about the effective index of the mode and the field profiles, as well as the mode_spec that was used in the solver. The effective index data and the field profile data is in the form of xarray DataArrays.

We can for example plot the real part of the effective index for all three modes as follows.

[8]:
fig, ax = plt.subplots(1)
n_eff = mode_data.n_eff # real part of the effective mode index
n_eff.plot.line(x='f');
../_images/notebooks_ModeSolver_15_0.png

The raw data can also be accessed.

[9]:
n_complex = mode_data.n_complex # complex effective index as a DataArray
n_eff = mode_data.n_eff.values  # real part of the effective index as numpy array
k_eff = mode_data.k_eff.values  # imag part of the effective index as numpy array

print(f'first mode effective index at freq0: n_eff = {n_eff[f0_ind, 0]:.2f}, k_eff = {k_eff[f0_ind, 0]:.2e}')
first mode effective index at freq0: n_eff = 1.77, k_eff = 0.00e+00

The fields stored in mode_data can be visualized using in-built xarray methods.

[10]:
f, (ax1, ax2) = plt.subplots(1, 2, tight_layout=True, figsize=(10, 3))
abs(mode_data.Ex.isel(mode_index=0, f=f0_ind)).plot(x='x', y='z', ax=ax1, cmap="magma")
abs(mode_data.Ez.isel(mode_index=0, f=f0_ind)).plot(x='x', y='z', ax=ax2, cmap='magma')

ax1.set_title('|Ex(x, y)|')
ax1.set_aspect("equal")
ax2.set_title('|Ez(x, y)|')
ax2.set_aspect("equal")
plt.show()
../_images/notebooks_ModeSolver_19_0.png

Alternatively, we can use the in-built plot_field method of mode_data, which also allows us to overlay the structures in the simulation. The image also looks slightly different because we have set robust=True by default, which scales the colorbar to between the 2nd and 98th percentile of the data.

[15]:
f, (ax1, ax2) = plt.subplots(1, 2, tight_layout=True, figsize=(10, 3))
mode_solver.plot_field("Ex", "abs", mode_index=0, f=freq0, ax=ax1)
mode_solver.plot_field("Ez", "abs", mode_index=0, f=freq0, ax=ax2)
plt.show()
../_images/notebooks_ModeSolver_21_0.png

Choosing the mode of interest#

We can also look at the other modes that were computed.

[16]:
mode_index = 1
f, (ax1, ax2) = plt.subplots(1, 2, tight_layout=True, figsize=(10, 3))
mode_solver.plot_field("Ex", "abs", mode_index=mode_index, f=freq0, ax=ax1)
mode_solver.plot_field("Ez", "abs", mode_index=mode_index, f=freq0, ax=ax2);
plt.show()
../_images/notebooks_ModeSolver_23_0.png

This looks like an Ez-dominant mode. Finally, next-order mode has mixed polarization.

[17]:
mode_index = 2
f, (ax1, ax2) = plt.subplots(1, 2, tight_layout=True, figsize=(10, 3))
mode_solver.plot_field("Ex", "abs", mode_index=mode_index, f=freq0, ax=ax1)
mode_solver.plot_field("Ez", "abs", mode_index=mode_index, f=freq0, ax=ax2);
plt.show()
../_images/notebooks_ModeSolver_25_0.png

Exporting Results#

This looks promising!

Now we can choose the mode specifications to use in our mode source and mode monitors. These can be created separately, can be exported directly from the mode solver, for example:

[18]:
# Makes a modal source with geometry of `plane` with modes specified by `mode_spec` and a selected `mode_index`
source_time = td.GaussianPulse(freq0=freq0, fwidth=fwidth)
mode_src = mode_solver.to_source(mode_index=2, source_time=source_time, direction='-')

# Makes a mode monitor with geometry of `plane`.
mode_mon = mode_solver.to_monitor(name='mode', freqs=freqs)
# Offset the monitor along the propagation direction
mode_mon = mode_mon.copy(update=dict(center = (0, -2, 0)))
[19]:
# In-plane field monitor, slightly offset along x
monitor = td.FieldMonitor(
    center=(0, 0, 0.1),
    size=(td.inf, td.inf, 0),
    freqs=[freq0],
    name='field'
)

sim = td.Simulation(
    size=(Lx, Ly, Lz),
    grid_spec=grid_spec,
    run_time=run_time,
    boundary_spec=td.BoundarySpec.all_sides(boundary=td.PML()),
    structures=[waveguide],
    sources=[mode_src],
    monitors=[monitor, mode_mon]
)

sim.plot(z=0);
../_images/notebooks_ModeSolver_28_0.png
[20]:
job = web.Job(simulation=sim, task_name='mode_simulation')
sim_data = job.run(path='data/simulation_data.hdf5')
 monitor_data.hdf5 ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╸━━━━━━━ 81.7%1.3/1.6 MB1.7 MB/s0:00:01
[09:17:19] INFO     loading SimulationData from data/simulation_data.hdf5                             webapi.py:397

We can now plot the in-plane field and the modal amplitudes. Since we injected mode 2 and we just have a straight waveguide, all the power recorded by the modal monitor is in mode 2, going backwards.

[23]:
fig, ax = plt.subplots(1, 2, figsize=(10, 4))
sim_data.plot_field("field", "Ez", f=freq0, ax=ax[0])
sim_data['mode'].amps.sel(direction='-').abs.plot.line(x='f', ax=ax[1]);
../_images/notebooks_ModeSolver_31_0.png

Storing server-side computed modes#

We can also use a ModeSolverMonitor to store the modes as they are computed server-side. This is illustrated below. We will also request in the mode specification that the modes are filtered by their tm polarization. In this particular simulation, TM refers to Ez polarization. The effect of the filtering is that modes with a tm polarization fraction larger than or equal to 0.5 will come first in the list of modes (while still ordered by decreasing effective index). After that, the set of predominantly te-polarized modes (tm fraction < 0.5) follow.

[26]:
mode_spec = mode_spec.copy(update=dict(filter_pol = "tm"))

# Update mode source to use the highest-tm-fraction mode
mode_src = mode_src.copy(update=dict(mode_spec = mode_spec))
mode_src = mode_src.copy(update=dict(mode_index = 0))

# Update mode monitor to use the tm_fraction ordered mode_spec
mode_mon = mode_mon.copy(update=dict(mode_spec = mode_spec))

# New monitor to record the modes computed at the mode decomposition monitor location
mode_solver_mon = td.ModeSolverMonitor(
    center=mode_mon.center,
    size=mode_mon.size,
    freqs=mode_mon.freqs,
    mode_spec=mode_spec,
    name="mode_solver"
)

sim = td.Simulation(
    size=(Lx, Ly, Lz),
    grid_spec=grid_spec,
    run_time=run_time,
    boundary_spec=td.BoundarySpec.all_sides(boundary=td.PML()),
    structures=[waveguide],
    sources=[mode_src],
    monitors=[monitor, mode_mon, mode_solver_mon]
)
[27]:
job = web.Job(simulation=sim, task_name='mode_simulation')
sim_data = job.run(path='data/simulation_data.hdf5')
 monitor_data.hdf5 ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╺━ 95.2%10.4/11.0 MB2.5 MB/s0:00:01
[09:20:13] INFO     loading SimulationData from data/simulation_data.hdf5                             webapi.py:397

Note the different ordering of the recorded modes compared to what we saw above, with the fundamental TM mode coming first.

[28]:
fig, ax = plt.subplots(1)
n_eff = sim_data["mode"].n_eff # real part of the effective mode index
n_eff.plot.line(x='f');
../_images/notebooks_ModeSolver_36_0.png

Now the fundamental Ez-polarized mode is injected, and as before it is the only one that the mode monitor records any intensity in.

[31]:
fig, ax = plt.subplots(1, 2, figsize=(10, 4))
sim_data.plot_field("field", "Ez", f=freq0, ax=ax[0])
sim_data['mode'].amps.sel(direction='-').abs.plot.line(x='f', ax=ax[1]);
../_images/notebooks_ModeSolver_38_0.png

We can also have a look at the mode fields stored in the ModeFieldMonitor either directly using xarray methods as above, or using the Tidy3D SimulationData in-built field plotting. Note that here we have to specificall

[32]:
fig, ax = plt.subplots(1, 2, figsize=(12, 4))
sim_data.plot_field("mode_solver", "Ex", f=freq0, val="abs", mode_index=0, ax=ax[0])
sim_data.plot_field("mode_solver", "Ez", f=freq0, val="abs", mode_index=0, ax=ax[1]);
../_images/notebooks_ModeSolver_40_0.png

Notes / Considerations#

  • This mode solver runs locally, which means it does not require credits to run.

  • It also means that the mode solver does not use subpixel-smoothening, even if this is specified in the simulation. On the other hand, when the modes are stored in a ModeSolverMonitor during a simulation run, the subpixel averaging is applied. Therefore, the latter results may not perfectly match those from a local run, and are more accurate.

  • Symmetries are applied if they are defined in the simulation and the mode plane center sits on the simulation center.

[ ]: