MPB mode-solver#
MPB is a free open source software to compute:
electro-magnetic modes
band structures
supported by a waveguide with periodic boundaries.
Find modes waveguide#
Lets find the modes supported by a waveguide for a particular waveguide geometry and wavelength.
A waveguide is like a pipe to guide the light and is made of a higher refractive index core core_material surrounded by a lower refractive index cladding clad_material
__________________________
|
|
| width
| <---------->
| ___________ _ _ _
| | | |
sz|_____| |_______|
| | core_thickness
|slab_thickness |
|_________________________|
|
|
|__________________________
<------------------------>
sy
Silicon is yellow and opaque at visible wavelengths (380 to 700nm). This is the reason why CMOS cameras can be made of Silicon.
At Infra-red wavelengths used for communications (1300 or 1550nm) Silicon is transparent and has a high refractive index 3.47. So making a Silicon waveguide is quite easy, where the Silicon is the guiding material, and Silicon oxide n=1.45 makes a great low index material for the cladding of the waveguide.
This video explains how Silicon Photonic waveguides guide light in Photonic integrated circuits.
Strip waveguides#
Strip waveguides are fully etch and don’t have a slab. slab_thickness = 0
__________________________
|
|
| width
| <---------->
| ___________ _ _ _
| | | |
sz| | | |
| | core_material | core_thickness
| | | |
| |___________| _ _ _|
|
| clad_material
|__________________________
<------------------------>
sy
import matplotlib.pyplot as plt
import meep as mp
import numpy as np
import gplugins.modes as gm
Using MPI version 4.1, 1 processes
2024-05-10 10:29:38.367 | INFO | gplugins.gmeep:<module>:39 - Meep '1.28.0' installed at ['/home/runner/micromamba/lib/python3.11/site-packages/meep']
modes = gm.find_modes_waveguide(
parity=mp.NO_PARITY,
core_width=0.4,
core_material=3.47,
clad_material=1.44,
core_thickness=0.22,
resolution=40,
sy=3,
sz=3,
nmodes=4,
)
m1 = modes[1]
m2 = modes[2]
m3 = modes[3]
As you can see the refractive index goes from 1.44 SiO2 Silicon dioxide to 3.47 Silicon.
m1.plot_eps()
m1.neff
2.2179432568410284
m1.plot_ey()
m2.plot_e_all()
m1.plot_e()
As you can see the first order mode has most power in y-direction Ey. This type of mode is called TE (transverse-electric)
On the other hand the second order mode has most of the light in the Ex. This mode is called TM (transverse-magnetic)
m2.plot_e_all()
m3.plot_e() # not guided
m1.neff
2.2179432568410284
m2.neff
1.6821915241555887
# the third mode does not propagate and its neff is below the cladding index
m3.neff
1.4306968409274778
Sidewall angle#
You can also specify the sidewall angle.
modes = gm.find_modes_waveguide(
parity=mp.NO_PARITY,
core_width=0.4,
core_material=3.47,
clad_material=1.44,
core_thickness=0.22,
resolution=40,
sidewall_angle=10,
)
m1 = modes[1]
m2 = modes[2]
m3 = modes[3]
m1.plot_eps()
modes = gm.find_modes_waveguide(
parity=mp.NO_PARITY,
core_width=0.4,
core_material=3.47,
clad_material=1.44,
core_thickness=0.22,
resolution=60,
sidewall_angle=10,
slab_thickness=90e-3,
)
m1 = modes[1]
m2 = modes[2]
m3 = modes[3]
m1.plot_eps()
Rib waveguides#
Rib waveguides have a slab (not fully etched)
modes = gm.find_modes_waveguide(
mode_number=1, nmodes=2, slab_thickness=90e-3, resolution=40
)
m1 = modes[1]
m2 = modes[2]
m1.plot_eps()
m1.plot_e_all()
Symmetries#
You can exploit symmetries to reduce computation time as well as finding only (TE or TM) modes
MPB assumes propagation in the X direction
TE: mp.ODD_Y + mp.EVEN_Z
TM: mp.EVEN+Y + mp.ODD_Z, all energy in z component
TM: mp.ODD_Y + mp.EVEN_Z#
You can define an even Y parity to find only the TM modes
modes = gm.find_modes_waveguide(
mode_number=1,
parity=mp.EVEN_Y + mp.ODD_Z,
nmodes=2,
core_width=1.0,
core_material=3.47,
clad_material=1.44,
core_thickness=0.22,
resolution=32,
sy=6,
sz=6,
)
m1 = modes[1]
m2 = modes[2]
m1.plot_e()
ODD_Y (TE)#
modes = gm.find_modes_waveguide(
mode_number=1,
parity=mp.ODD_Y,
nmodes=2,
core_width=0.20,
core_material=3.47,
clad_material=1.44,
core_thickness=0.22,
resolution=20,
sy=5,
sz=5,
)
m1 = modes[1]
m2 = modes[2]
m1.plot_e()
Sweep waveguide width#
Strip#
df = gm.find_neff_vs_width(filepath="data/mpb_neff_vs_width.csv")
df
| 1 | 2 | 3 | 4 | width | |
|---|---|---|---|---|---|
| 0 | 1.499770 | 1.488664 | 1.328191 | 1.316624 | 0.200000 |
| 1 | 1.683443 | 1.556811 | 1.384970 | 1.358053 | 0.272727 |
| 2 | 2.031569 | 1.632182 | 1.406476 | 1.375044 | 0.345455 |
| 3 | 2.261402 | 1.692583 | 1.413844 | 1.400262 | 0.418182 |
| 4 | 2.410539 | 1.744925 | 1.418374 | 1.418373 | 0.490909 |
| 5 | 2.508256 | 1.788094 | 1.580987 | 1.422537 | 0.563636 |
| 6 | 2.577841 | 1.823844 | 1.767065 | 1.428408 | 0.636364 |
| 7 | 2.627689 | 1.958723 | 1.853650 | 1.438660 | 0.709091 |
| 8 | 2.665757 | 2.122754 | 1.880209 | 1.489468 | 0.781818 |
| 9 | 2.691444 | 2.236792 | 1.898304 | 1.540170 | 0.854545 |
| 10 | 2.711755 | 2.326842 | 1.913680 | 1.649241 | 0.927273 |
| 11 | 2.727891 | 2.397685 | 1.926424 | 1.792733 | 1.000000 |
gm.plot_neff_vs_width(df)
Rib#
modes = gm.find_modes_waveguide(
core_width=0.4,
core_material=3.47,
clad_material=1.44,
core_thickness=220e-3,
resolution=20,
sz=6,
sy=6,
nmodes=4,
slab_thickness=90e-3,
)
m1 = modes[1]
m2 = modes[2]
m3 = modes[3]
m1.plot_eps()
m1.neff
2.446049805846579
m1.plot_e()
m1.neff
2.446049805846579
m2.plot_e()
m2.neff
2.059679078398959
df = gm.find_neff_vs_width(
slab_thickness=90e-3, filepath="data/mpb_neff_vs_width_rib.csv"
)
gm.plot_neff_vs_width(df)
Nitride#
modes = gm.find_modes_waveguide(
core_width=1.0,
core_material=2.0,
clad_material=1.44,
core_thickness=400e-3,
sz=6,
sy=10,
nmodes=4,
resolution=10,
)
m1 = modes[1]
m2 = modes[2]
m3 = modes[3]
m1.plot_eps()
m1.plot_ey()
m1.plot_e_all()
m2.plot_ex()
m3.plot_ey()
df = gm.find_neff_vs_width(
width1=0.5,
width2=1.2,
core_thickness=0.4,
core_material=2.0,
sy=10.0,
resolution=15,
filepath="data/mpb_neff_vs_width_nitride.csv",
)
gm.plot_neff_vs_width(df)
Dispersion#
To get the effective index we only need to compute the mode propagation constant at a single frequency.
However, to compute the dispersion (group delay) we need to compute the effective index for at least 3 wavelengths.
The effective index neff relates to the speed of the phase evolution of the light, while the group index ng relates to the group velocity of the light.
To compute the resonances in MZI interferometers or ring resonators you need to use ng
help(gm.find_mode_dispersion)
Help on function find_mode_dispersion in module gplugins.modes.find_mode_dispersion:
find_mode_dispersion(wavelength: 'float' = 1.55, wavelength_step: 'float' = 0.01, core: 'str' = 'Si', clad: 'str' = 'SiO2', mode_number: 'int' = 1, **kwargs) -> 'Mode'
Returns Mode with correct dispersion (ng).
group index comes from a finite difference approximation at 3 wavelengths
Args:
wavelength: center wavelength (um).
wavelength_step: in um.
core: core material name.
clad: clad material name.
mode_number: mode index to compute (1: fundamental mode).
Keyword Args:
core_thickness: wg height (um).
sx: supercell width (um).
sy: supercell height (um).
resolution: (pixels/um).
wavelength: wavelength.
num_bands: mode order.
plot: if True plots mode.
logscale: plots in logscale.
plotH: plot magnetic field.
cache: path to save the modes.
polarization: prefix when saving the modes.
parity: symmetries mp.ODD_Y mp.EVEN_X for TE, mp.EVEN_Y for TM.
m = gm.find_mode_dispersion()
m.ng
4.2762565439336155
Convergence tests#
Before launching a set of simulations you need to make sure you have the correct simulation settings:
resolution: resolution
sx: Size of the simulation region in the x-direction (default=4.0)
sy: Size of the simulation region in the y-direction (default=4.0)
resolutions = np.linspace(10, 50, 5)
neffs = []
for resolution in resolutions:
modes = gm.find_modes_waveguide(
core_width=0.5,
core_material=3.5,
clad_material=1.44,
core_thickness=0.22,
resolution=resolution,
)
mode = modes[1]
neffs.append(mode.neff)
plt.plot(resolutions, neffs, "o-")
plt.ylabel("neff")
plt.xlabel("resolution (pixels/um)")
Text(0.5, 0, 'resolution (pixels/um)')
szs = np.linspace(4, 6, 6)
neffs = []
for sz in szs:
modes = gm.find_modes_waveguide(
core_width=0.5,
core_material=3.5,
clad_material=1.44,
core_thickness=0.22,
resolution=20,
sz=sz,
)
mode = modes[1]
neffs.append(mode.neff)
plt.plot(szs, neffs, "o-")
plt.ylabel("neff")
plt.xlabel("simulation size in z(um)")
Text(0.5, 0, 'simulation size in z(um)')
sys = np.linspace(2, 6, 6)
neffs = []
for sy in sys:
modes = gm.find_modes_waveguide(
core_width=0.5,
core_material=3.5,
clad_material=1.44,
core_thickness=0.22,
resolution=20,
sy=sy,
)
mode = modes[1]
neffs.append(mode.neff)
plt.plot(sys, neffs, "o-")
plt.ylabel("neff")
plt.xlabel("simulation size in y (um)")
Text(0.5, 0, 'simulation size in y (um)')
Find modes coupler#
When two waveguides are close to each other, they support modes that travel with different index (speed). One of the modes is an even mode, while the other one is an odd mode.
Light will couple from one waveguide to another because the even and odd modes travel at different speeds and they interfere with each other. Creating a periodically back and forth coupling between both waveguides.
Depending on the length of the coupling region and the gap there will be a different percentage of the light coupled from one to another
_____________________________________________________
|
|
| widths[0] widths[1]
| <----------> gaps[0] <---------->
| ___________ <-------------> ___________ _
| | | | | |
sz|_____| |_______________| |_____|
| core_material | core_thickness
|slab_thickness nslab |
|___________________________________________________|
|
|<---> <--->
|ymargin clad_material ymargin
|____________________________________________________
<--------------------------------------------------->
sy
modes = gm.find_modes_coupler(
core_widths=(0.5, 0.5),
gaps=(0.2,),
core_material=3.47,
clad_material=1.44,
core_thickness=0.22,
resolution=20,
sz=6,
nmodes=4,
)
m1 = modes[1]
m2 = modes[2]
m3 = modes[3]
m1.plot_eps()
m1.plot_ey() # even mode
m2.plot_ey() # odd mode
Find coupling vs gap#
gm.find_coupling_vs_gap?
df = gm.coupler.find_coupling_vs_gap(
gap1=0.2,
gap2=0.4,
steps=12,
nmodes=4,
wavelength=1.55,
filepath="data/mpb_find_coupling_vs_gap_strip.csv",
)
plt.title("strip 500x200 coupling")
gm.plot_coupling_vs_gap(df)
df = gm.coupler.find_coupling_vs_gap_nitride(
filepath="data/mpb_find_coupling_vs_gap_nitride.csv"
)
plt.title("nitride 1000x400 nitride")
gm.plot_coupling_vs_gap(df)
ne = []
no = []
gaps = [0.2, 0.25, 0.3]
for gap in gaps:
modes = gm.find_modes_coupler(
core_widths=(0.5, 0.5),
gaps=(gap,),
core_material=3.47,
clad_material=1.44,
core_thickness=0.22,
resolution=20,
sz=6,
nmodes=4,
)
ne.append(modes[1].neff)
no.append(modes[2].neff)
def coupling_length(
neff1: float,
neff2: float,
power_ratio: float = 1.0,
wavelength: float = 1.55,
) -> float:
"""Returns the coupling length (um) of the directional coupler
to achieve power_ratio.
Args:
neff1: even supermode of the directional coupler.
neff2: odd supermode of the directional coupler.
power_ratio: p2/p1, where 1 means 100% power transfer.
wavelength: in um.
"""
dneff = (neff1 - neff2).real
return wavelength / (np.pi * dneff) * np.arcsin(np.sqrt(power_ratio))
lc = [
coupling_length(neff1=neff1, neff2=neff2) for gap, neff1, neff2 in zip(gaps, ne, no)
]
plt.plot(gaps, lc, ".-")
plt.ylabel("100% coupling length (um)")
plt.xlabel("gap (um)")
Text(0.5, 0, 'gap (um)')
