Reflectance Performance of the W-coat on a Glass Substrate¶
This notebook demonstrates how to calculate the reflectance of a multilayer thin film stack (W-coat) deposited on a glass substrate. The calculation is performed for two types of polarizations, s-polarization and p-polarization, over a wavelength range of 500 nm to 700 nm and incident angles from 0° to 70°.
Introduction¶
We will use the Transfer Matrix Method (TMM) to analyze the reflection behavior of this multilayer thin film. The W-coat consists of two layers made from Magnesium Fluoride (MgF₂) and Yttrium Oxide (Y₂O₃). These layers are stacked on a glass substrate (Silicon Dioxide, SiO₂), and the incident medium is air.
In this example, the calculation is done for two polarizations:
s-polarization: Electric field perpendicular to the plane of incidence.
p-polarization: Electric field parallel to the plane of incidence.
We will calculate the reflectance as a function of wavelength and angle of incidence for each polarization.
Materials and Layer Thickness¶
We are considering the following material stack:
Air as the incident medium.
MgF₂ (Magnesium Fluoride) with a thickness of 93 nm.
Y₂O₃ (Yttrium Oxide) with a thickness of 63 nm.
SiO₂ (Silicon Dioxide) as the substrate.
These materials are well-suited for anti-reflection coatings and optical filters due to their refractive indices.
Python Implementation¶
In this section, we implement the TMM algorithm using the tmmax library, which is designed for high-performance multilayer simulations with vectorized operations using the jax library.
[ ]:
import jax.numpy as jnp
from tmmax.tmm import tmm
# Define the materials and their thicknesses
material_list = ["Air", "MgF2", "Y2O3", "SiO2"]
thickness_list = jnp.array([93e-9, 63e-9])
# Define the wavelength range (500 nm to 700 nm) and angles of incidence (0° to 70°)
wavelength_arr = jnp.linspace(500e-9, 700e-9, 1000)
angle_of_incidences = jnp.linspace(0, (70*jnp.pi/180), 1000)
# Calculate reflectance for s-polarization
result_s = tmm(material_list=material_list, thickness_list=thickness_list,
wavelength_arr=wavelength_arr, angle_of_incidences=angle_of_incidences,
polarization="s")
# Calculate reflectance for p-polarization
result_p = tmm(material_list=material_list, thickness_list=thickness_list,
wavelength_arr=wavelength_arr, angle_of_incidences=angle_of_incidences,
polarization="p")
Visualization of Results¶
We will now visualize the reflectance for both s-polarization and p-polarization as a function of wavelength and angle of incidence. The color map will represent the reflectance percentage.
[ ]:
import matplotlib.pyplot as plt
from mpl_toolkits.axes_grid1 import ImageGrid
# Set up figure and image grid
fig = plt.figure(figsize=(10, 5))
grid = ImageGrid(fig, 111,
nrows_ncols=(1,2),
axes_pad=0.65,
share_all=True,
cbar_location="right",
cbar_mode="single",
cbar_size="5%",
cbar_pad=0.25,
)
# Add data to image grid
i = 0
for ax in grid:
if i == 0:
im = ax.imshow(result_s[0]*100, cmap='Spectral', aspect=200/70,extent = [500, 700, 0, 70])
ax.set_title("s-polarization")
if i ==1:
im = ax.imshow(result_p[0]*100, cmap='Spectral', aspect=200/70,extent = [500, 700, 0, 70])
ax.set_title("p-polarization")
ax.set_xlabel("Wavelength (nm)")
ax.set_ylabel("Incident angle ($^o$)")
i += 1
# Colorbar
ax.cax.colorbar(im, label='Reflectance (%)')
plt.savefig("w_coat_reflectance_s_p.png", dpi=600)
plt.show()
Discussion of Reflectance Results¶
s-Polarization¶
The first heatmap (on the left) represents the reflectance of the thin-film stack for s-polarized light (electric field perpendicular to the plane of incidence). The key observations from this plot include:
Reflectance Trends:
At lower wavelengths (~500 nm), the reflectance is higher for near-normal incidence (small angles) and gradually decreases as the angle of incidence increases. This is visually apparent by the transition from a light green at the lower left corner to darker colors at higher angles.
For wavelengths above ~550 nm, the reflectance remains relatively low across all angles of incidence. This is indicated by the uniformity in the darker red and purple regions of the heatmap.
Angle Dependence:
As the angle increases, the reflectance generally rises for wavelengths below ~600 nm. This behavior can be seen in the color gradient from the blue/green regions at small angles to the red/maroon shades at higher angles.
The most significant variation occurs for wavelengths in the range of 500-525 nm, where a sharp color transition occurs, indicating higher reflectance at lower wavelengths.
p-Polarization¶
The second heatmap (on the right) shows the reflectance for p-polarized light (electric field parallel to the plane of incidence). The trends for p-polarization differ from those of s-polarization in several ways:
Reflectance Trends:
Similar to s-polarization, the reflectance is higher at shorter wavelengths and for near-normal incidence. However, p-polarization shows a slightly higher overall reflectance in the shorter wavelength region (500-525 nm), as seen in the more pronounced blue and green regions.
Above ~600 nm, the reflectance remains consistently low across the entire angle range, much like in s-polarization, which is reflected in the red-to-maroon shading.
Angle Dependence:
For p-polarized light, the reflectance behavior at small angles and shorter wavelengths (500-525 nm) shows a slightly steeper rise compared to s-polarization. The blue-to-green transition occurs at smaller angles in this case, indicating a more pronounced increase in reflectance at these angles.
As the angle approaches 70°, the reflectance tends to stabilize and reduce, particularly for longer wavelengths.
Comparison of s- and p-Polarization Reflectance¶
At shorter wavelengths (500-525 nm), both s- and p-polarization reflectance exhibit higher values at normal incidence and increase with the angle of incidence. However, p-polarization shows a steeper increase compared to s-polarization at these wavelengths.
For wavelengths above ~600 nm, the reflectance remains consistently low for both polarizations across all angles of incidence, though the p-polarization shows marginally higher values at lower angles.
Transmittance Results¶
[ ]:
import matplotlib.pyplot as plt
from mpl_toolkits.axes_grid1 import ImageGrid
# Set up figure and image grid
fig = plt.figure(figsize=(10, 5))
grid = ImageGrid(fig, 111,
nrows_ncols=(1,2),
axes_pad=0.65,
share_all=True,
cbar_location="right",
cbar_mode="single",
cbar_size="5%",
cbar_pad=0.25,
)
# Add data to image grid
i = 0
for ax in grid:
if i == 0:
im = ax.imshow(result_s[1]*100, cmap='Spectral', aspect=200/70,extent = [500, 700, 0, 70])
ax.set_title("s-polarization")
if i ==1:
im = ax.imshow(result_p[1]*100, cmap='Spectral', aspect=200/70,extent = [500, 700, 0, 70])
ax.set_title("p-polarization")
ax.set_xlabel("Wavelength (nm)")
ax.set_ylabel("Incident angle ($^o$)")
i += 1
# Colorbar
ax.cax.colorbar(im, label='Transmittance (%)')
plt.savefig("w_coat_transmittance_s_p.png", dpi=600)
plt.show()
Absorbance Results¶
[ ]:
import matplotlib.pyplot as plt
from mpl_toolkits.axes_grid1 import ImageGrid
# Set up figure and image grid
fig = plt.figure(figsize=(10, 5))
grid = ImageGrid(fig, 111,
nrows_ncols=(1,2),
axes_pad=0.65,
share_all=True,
cbar_location="right",
cbar_mode="single",
cbar_size="5%",
cbar_pad=0.25,
)
# Add data to image grid
i = 0
for ax in grid:
if i == 0:
im = ax.imshow((1 -result_s[0] - result_s[1])*100, cmap='Spectral', aspect=200/70,extent = [500, 700, 0, 70])
ax.set_title("s-polarization")
if i ==1:
im = ax.imshow((1 -result_p[0] - result_p[1])*100, cmap='Spectral', aspect=200/70,extent = [500, 700, 0, 70])
ax.set_title("p-polarization")
ax.set_xlabel("Wavelength (nm)")
ax.set_ylabel("Incident angle ($^o$)")
i += 1
# Colorbar
ax.cax.colorbar(im, label='Absorbance (%)')
plt.savefig("w_coat_absorbance_s_p.png", dpi=600)
plt.show()
