Modeling and Electromagnetic simulation#
import time
import matplotlib.pyplot as plt
import numpy as np
import meent
pol = 0 # 0: TE, 1: TM
n_top = 1 # n_superstrate
n_bot = 1 # n_substrate
theta = 20 * np.pi / 180
phi = 50 * np.pi / 180
wavelength = 900
thickness = [500]
period = [1000]
fto = [30]
type_complex = np.complex128
1.1 Modeling#
ucell (unit cell)#
ucell (unit cell) is a 3-dimensional array which has refractive index elements. It supports complex index, 2-dimensional grating and multi-layered structures.
The order of dimensions: Z Y X.
______________________
|\ \
| \ \ Y
| \ \
|\ \_____________________\
| \ | Layer 1 |
\ \|_____________________|
\ | | Z
\ | Layer 2 |
\|_____________________|
X
Z: Stack of the layers (1st axis of ucell)
In z-direction, layers (2D arrays) are stacked from top to bottom.
For example, ucell[0] is Layer 1 in the figure (2D array on the top) and ucell[1] is the next layer below.
Y: Depth of the structure (2nd axis of ucell)
2 dimensional grating is implemented on this and the last axes.
For 1 dimensional case, this is not used and fixed to 1.
X: Width of the structure (3rd axis of ucell)
This is the default axis. 1 dimensional grating is implemented on this axis.
ucell examples#
ucell_1d_s = np.array([
[
[0, 1, 0, 1, 1, 0, 1, 0, 1, 1, ],
],
]) * 4 + 1 # refractive index
1D grating, single layer
ucell_1d_m = np.array([
[
[0, 1, 0, 1, 1, 0, 1, 0, 1, 1, ],
],
[
[1, 1, 1, 1, 1, 0, 1, 0, 1, 0, ],
],
[
[0, 0, 0, 0, 0, 0, 1, 0, 1, 1, ],
],
]) * 4 + 1 # refractive index
1D grating, multi layer
ucell_2d_s = np.array([
[
[0, 1, 0, 1, 1, 0, 1, 0, 1, 1, ],
[1, 1, 1, 1, 1, 1, 1, 0, 1, 1, ],
],
]) * 4 + 1 # refractive index
2D grating, single layer
ucell_2d_m = np.array([
[
[0, 1, 0, 1, 1, 0, 1, 0, 1, 1, ],
[1, 1, 1, 1, 1, 1, 1, 0, 1, 1, ],
],
[
[1, 1, 0, 1, 1, 0, 1, 0, 2, 1, ],
[0, 1, 1, 1, 2, 4, 1, 0, 1, 1, ],
],
[
[0, 1, 0, 1, 1, 0, 1, 0, 1, 1, ],
[1, 1, 1, 2, 0, 1, 2, 0, 1, 1, ],
],
[
[0, 1, 0, 1, 1, 1, 1, 0, 1, 1, ],
[0, 1, 3, 1, 1, 1, 1, 1, 1, 1, ],
],
]) * 4 + 1 # refractive index
2D grating, multi layer
Wave#
$\theta$: angle of incidence#
Incidence normal
\\ |
\\ |
\\ 𝜃 (angle of incidence)
\\ |
\\ |
\\ |
\\| Z-axis
____________________________________ |
| Layer 1 | |
|____________________________________| |_____ X-axis
$\phi$: azimuthal (rotation) angle#
Plane of Incidence
/ |
/ |
/ |
/ |
/ |
/ |
/ |
/ /
/ /
/ / / /
/ / / 𝜙 (rotation angle) /
/ / -----/------ /
/ / / / / Y-axis
/ | / / / /
/_______|__________/_____________________/ / / /
| | / Layer 1 | / / /_____ X-axis
|_______|________/_______________________|/ / |
| | / Layer 2 | / |
|_______|______/_________________________|/ Z-axis
| /
In 2D (projection on XY-plane),
Y-axis Plane of Incidence
| //
| //
| 𝜙 // (rotation angle)
| //
| //
|//
|--------------- X-axis
|
|
1.2 EM Simulation#
mee = meent.call_mee(backend=0, pol=pol, n_top=n_top, n_bot=n_bot, theta=theta, phi=phi, fto=fto, wavelength=wavelength, period=period, ucell=ucell_1d_s, thickness=thickness, type_complex=type_complex)
call meent operator by meent.call_mee.
Here, backend can be selected with keyword backend
backend = 0 # Numpy backend
backend = 1 # JAX backend
backend = 2 # PyTorch backend
Diffraction Efficiency#
Diffraction#
Incidence Backward Diffraction
(Reflected)
||
|| -1th 0th +1th
|| order order order
|| \ | /
|| ... \ | / ...
|| \ | /
|| \ | / n_top:refractive index of superstrate
____________________________________
| Layer 1 |
|____________________________________|
. z-axis
. |
. |
____________________________________ |_____ x-axis
| Layer N |
|____________________________________|
n_bot:refractive index of substrate
/ | \
/ | \
... / | \ ...
/ | \
-2nd -1th 0th +1th +2th
order order order order order
Forward Diffraction
(Transmitted)
t0 = time.time()
de_ri, de_ti = mee.conv_solve()
print(f'time: ', time.time() - t0)
time: 0.11336421966552734
center = de_ri.shape[0] // 2
print('Diffraction Efficiency of Reflection:', np.round(de_ri[center-1:center+2], 3))
print('Diffraction Efficiency of Transmission:', np.round(de_ti[center-1:center+2], 3))
Diffraction Efficiency of Reflection: [[0. ]
[0.287]
[0. ]]
Diffraction Efficiency of Transmission: [[0. ]
[0.713]
[0. ]]
Field Distribution#
t0 = time.time()
field_cell = mee.calculate_field(res_z=100, res_y=1, res_x=100)
print(f'time: ', time.time() - t0)
time: 0.04631304740905762
ZX direction (Side View)#
fig, axes = plt.subplots(1, 3, figsize=(10,2))
if pol == 0: # TE
title = ['1D Ey', '1D Hx', '1D Hz', ]
else: # TM
title = ['1D Hy', '1D Ex', '1D Ez', ]
for ix in range(len(title)):
val = abs(field_cell[:, 0, :, ix]) ** 2
im = axes[ix].imshow(val, cmap='jet', aspect='auto')
# plt.clim(0, 2) # identical to caxis([-4,4]) in MATLAB
fig.colorbar(im, ax=axes[ix], shrink=1)
axes[ix].title.set_text(title[ix])
plt.show()
YZ Direction (Top View)#
fig, axes = plt.subplots(1, 3, figsize=(10,2))
if pol == 0: # TE
title = ['1D Ey', '1D Hx', '1D Hz', ]
else: # TM
title = ['1D Hy', '1D Ex', '1D Ez', ]
for ix in range(len(title)):
val = abs(field_cell[0, :, :, ix]) ** 2
im = axes[ix].imshow(val, cmap='jet', aspect='auto')
# plt.clim(0, 2) # identical to caxis([-4,4]) in MATLAB
fig.colorbar(im, ax=axes[ix], shrink=1)
axes[ix].title.set_text(title[ix])
plt.show()
1.3 Example: multilayer 2D#
grating_type = 2
fto = [10, 9]
thickness = [100, 200, 400, 245]
period = [1000, 2000]
ucell_2d_m = np.array([
[
[0, 1, 0, 1, 1, 0, 1, 0, 1, 1, ],
[1, 1, 1, 1, 1, 1, 1, 0, 1, 1, ],
],
[
[1, 1, 0, 1, 1, 0, 1, 0, 2, 1, ],
[0, 1, 1, 1, 2, 4, 1, 0, 1, 1, ],
],
[
[0, 1, 0, 1, 1, 0, 1, 0, 1, 1, ],
[1, 1, 1, 2, 0, 1, 2, 0, 1, 1, ],
],
[
[0, 1, 0, 1, 1, 1, 1, 0, 1, 1, ],
[0, 1, 3, 1, 1, 1, 1, 1, 1, 1, ],
],
]) * 4 + 1 # refractive index
mee = meent.call_mee(backend=0, pol=pol, n_top=n_top, n_bot=n_bot, theta=theta, phi=phi, fto=fto, wavelength=wavelength, period=period, ucell=ucell_2d_m, thickness=thickness, type_complex=type_complex)
t0 = time.time()
de_ri, de_ti = mee.conv_solve()
print(f'time: ', time.time() - t0)
time: 10.542928695678711
center = de_ri.shape[0] // 2
print('Diffraction Efficiency of Reflection:\n', np.round(de_ri[center-1:center+2, center-1:center+2], 3))
print('Diffraction Efficiency of Transmission:\n', np.round(de_ti[center-1:center+2, center-1:center+2], 3))
Diffraction Efficiency of Reflection:
[[0.028 0.056 0. ]
[0.183 0.036 0. ]
[0. 0. 0. ]]
Diffraction Efficiency of Transmission:
[[0.063 0.004 0. ]
[0.025 0.052 0. ]
[0. 0. 0. ]]
t0 = time.time()
field_cell = mee.calculate_field(res_z=40, res_y=40, res_x=40)
print(f'time: ', time.time() - t0)
time: 3.1625919342041016
ZX direction (Side View)#
fig, axes = plt.subplots(1, 3, figsize=(10, 2))
title = ['2D Ex', '2D Ey', '2D Ez', '2D Hx', '2D Hy', '2D Hz', ]
for ix in range(3):
val = abs(field_cell[:, 0, :, ix]) ** 2
im = axes[ix].imshow(val, cmap='jet', aspect='auto')
# plt.clim(0, 2) # identical to caxis([-4,4]) in MATLAB
fig.colorbar(im, ax=axes[ix], shrink=1)
axes[ix].title.set_text(title[ix])
plt.show()
fig, axes = plt.subplots(1, 3, figsize=(10, 2))
for ix in range(3, 6, 1):
val = abs(field_cell[:, 0, :, ix]) ** 2
im = axes[ix-3].imshow(val, cmap='jet', aspect='auto')
# plt.clim(0, 2) # identical to caxis([-4,4]) in MATLAB
fig.colorbar(im, ax=axes[ix-3], shrink=1)
axes[ix-3].title.set_text(title[ix])
plt.show()
YZ Direction (Top View)#
fig, axes = plt.subplots(1, 3, figsize=(10, 2))
title = ['2D Ex', '2D Ey', '2D Ez', '2D Hx', '2D Hy', '2D Hz', ]
for ix in range(3):
val = abs(field_cell[0, :, :, ix]) ** 2
im = axes[ix].imshow(val, cmap='jet', aspect='auto')
# plt.clim(0, 2) # identical to caxis([-4,4]) in MATLAB
fig.colorbar(im, ax=axes[ix], shrink=1)
axes[ix].title.set_text(title[ix])
plt.show()
fig, axes = plt.subplots(1, 3, figsize=(10, 2))
for ix in range(3, 6, 1):
val = abs(field_cell[0, :, :, ix]) ** 2
im = axes[ix-3].imshow(val, cmap='jet', aspect='auto')
# plt.clim(0, 2) # identical to caxis([-4,4]) in MATLAB
fig.colorbar(im, ax=axes[ix-3], shrink=1)
axes[ix-3].title.set_text(title[ix])
plt.show()
