Mode Solver
Contents
Mode Solver#
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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)
[16:32:42] WARNING No sources in simulation. simulation.py:406
<Figure size 432x288 with 1 Axes>
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');
[ <matplotlib.lines.Line2D object at 0x7fde85223460>, <matplotlib.lines.Line2D object at 0x7fde852234c0>, <matplotlib.lines.Line2D object at 0x7fde852235e0> ]
<Figure size 432x288 with 1 Axes>
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.fields can be visualized using in-built xarray methods.
[10]:
field_data = mode_data.fields
f, (ax1, ax2) = plt.subplots(1, 2, tight_layout=True, figsize=(10, 3))
abs(field_data.Ex.isel(mode_index=0, f=f0_ind)).plot(x='x', y='z', ax=ax1, cmap="magma")
abs(field_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()
<Figure size 720x216 with 4 Axes>
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.
[11]:
f, (ax1, ax2) = plt.subplots(1, 2, tight_layout=True, figsize=(10, 3))
mode_data.plot_field("Ex", "abs", mode_index=0, freq=freq0, ax=ax1)
mode_data.plot_field("Ez", "abs", mode_index=0, freq=freq0, ax=ax2)
plt.show()
<Figure size 720x216 with 4 Axes>
Choosing the mode of interest#
We can also look at the other modes that were computed.
[12]:
mode_index = 1
f, (ax1, ax2) = plt.subplots(1, 2, tight_layout=True, figsize=(10, 3))
mode_data.plot_field("Ex", "abs", mode_index=mode_index, freq=freq0, ax=ax1)
mode_data.plot_field("Ez", "abs", mode_index=mode_index, freq=freq0, ax=ax2);
plt.show()
<Figure size 720x216 with 4 Axes>
This looks like an Ez-dominant mode. Finally, next-order mode has mixed polarization.
[13]:
mode_index = 2
f, (ax1, ax2) = plt.subplots(1, 2, tight_layout=True, figsize=(10, 3))
mode_data.plot_field("Ex", "abs", mode_index=mode_index, freq=freq0, ax=ax1)
mode_data.plot_field("Ez", "abs", mode_index=mode_index, freq=freq0, ax=ax2);
plt.show()
<Figure size 720x216 with 4 Axes>
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:
[14]:
# 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)))
[15]:
# 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);
<AxesSubplot:title={'center':'cross section at z=0.00'}, xlabel='x', ylabel='y'>
<Figure size 432x288 with 1 Axes>
[16]:
job = web.Job(simulation=sim, task_name='mode_simulation')
sim_data = job.run(path='data/simulation_data.hdf5')
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.
[17]:
fig, ax = plt.subplots(1, 2, figsize=(10, 4))
sim_data.plot_field("field", "Ez", freq=freq0, ax=ax[0])
sim_data['mode']['amps'].sel(direction='-').abs.plot.line(x='f', ax=ax[1]);
[ <matplotlib.lines.Line2D object at 0x7fde852b6400>, <matplotlib.lines.Line2D object at 0x7fde852b6940>, <matplotlib.lines.Line2D object at 0x7fde852b6790> ]
<Figure size 720x288 with 3 Axes>
Storing server-side computed modes#
We can also use a ModeFieldMonitor 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.
[18]:
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.ModeFieldMonitor(
center=mode_mon.center,
size=mode_mon.size,
freqs=mode_mon.freqs,
mode_spec=mode_spec,
name="mode_fields"
)
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]
)
[19]:
job = web.Job(simulation=sim, task_name='mode_simulation')
sim_data = job.run(path='data/simulation_data.hdf5')
Note the different ordering of the recorded modes compared to what we saw above, with the fundamental TM mode coming first.
[20]:
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');
[ <matplotlib.lines.Line2D object at 0x7fde851a43d0>, <matplotlib.lines.Line2D object at 0x7fde84ed2af0>, <matplotlib.lines.Line2D object at 0x7fde84ed2d30> ]
<Figure size 432x288 with 1 Axes>
Now the fundamental Ez-polarized mode is injected, and as before it is the only one that the mode monitor records any intensity in.
[21]:
fig, ax = plt.subplots(1, 2, figsize=(10, 4))
sim_data.plot_field("field", "Ez", freq=freq0, ax=ax[0])
sim_data['mode']['amps'].sel(direction='-').abs.plot.line(x='f', ax=ax[1]);
[ <matplotlib.lines.Line2D object at 0x7fde83abc880>, <matplotlib.lines.Line2D object at 0x7fde83abc640>, <matplotlib.lines.Line2D object at 0x7fde83abcd60> ]
<Figure size 720x288 with 3 Axes>
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
[22]:
fig, ax = plt.subplots(1, 2, figsize=(12, 4))
sim_data.plot_field("mode_fields", "Ex", freq=freq0, val="abs", mode_index=0, ax=ax[0])
sim_data.plot_field("mode_fields", "Ez", freq=freq0, val="abs", mode_index=0, ax=ax[1]);
<AxesSubplot:title={'center':'cross section at y=-2.00'}, xlabel='x', ylabel='z'>
<Figure size 864x288 with 4 Axes>
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
ModeFieldMonitorduring 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.