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Coatings

Introduction to Coatings

Model thin-film layers that control reflection and transmission.

IntermediateCoatings & PolarizationNumPy backend12 min read

Introduction

This tutorial demonstrates how to apply coatings on surfaces in Optiland and how to assess their impact. Coatings impact the nature of light as it propagates through an optical system. For example, uncoated optics can result in significant light loss in a lens, especially if many surfaces are present.

In this tutorial, we will analyze two coating types:

  • "Simple" coatings
  • "Fresnel" coatings

The details of each coating type will be explained in its respective section.

Core concepts used

coatings.SimpleCoating(transmittance=0.5)
Assigns a constant intensity loss to a surface, regardless of angle or wavelength. Useful for quick 'budgeting'.
updater.set_fresnel_coating(index)
Uses the Fresnel equations to calculate reflectance based on the refractive index jump at the interface.
rays.i
The intensity attribute of a ray. It starts at 1.0 and decreases as the ray encounters reflective or absorbing surfaces.
PolarizationState(is_polarized=False)
Configures the system to model unpolarized light, essential for accurate Fresnel calculations.

Step-by-step build

1

Import matplotlib, numpy, and optiland coatings

python
import matplotlib.pyplot as plt
import numpy as np

from optiland import coatings, optic
from optiland.rays import PolarizationState
2

Define the uncoated doublet lens class

Let's first define a simple doublet without coatings.

python
class Doublet(optic.Optic):
 def __init__(self):
     super().__init__()

     self.add_surface(index=0, radius=np.inf, thickness=np.inf)
     self.add_surface(
         index=1,
         radius=29.32908,
         thickness=0.7,
         material="N-BK7",
         is_stop=True,
     )
     self.add_surface(index=2, radius=-20.06842, thickness=0.032)
     self.add_surface(
         index=3,
         radius=-20.08770,
         thickness=0.5780,
         material=("SF2", "schott"),
     )
     self.add_surface(index=4, radius=-66.54774, thickness=47.3562)
     self.add_surface(index=5)

     self.set_aperture(aperture_type="imageFNO", value=8.0)

     self.set_field_type(field_type="angle")
     self.add_field(y=0.0)
     self.add_field(y=0.7)
     self.add_field(y=1.0)

     self.add_wavelength(value=0.4861)
     self.add_wavelength(value=0.5876, is_primary=True)
     self.add_wavelength(value=0.6563)

     self.image_solve()
3

Instantiate and draw the uncoated doublet

python
lens0 = Doublet()
lens0.draw(figsize=(14, 2))
Step
4

Trace rays and plot the intensity distribution

We assess the intensity transmitted through the system by first tracing rays through the doublet, then analyzing the resulting ray intensity values at the image plane. This tutorial uses a similar approach to that shown in Tutorial 2a.

python
# trace rays through on-axis field point
rays = lens0.trace(Hx=0, Hy=0, wavelength=0.55, num_rays=256, distribution="uniform")

# take intensities of rays after ray tracing
intensity = rays.i

# compute average intensity per ray
avg_intensity = np.mean(intensity)

print(f"Average intensity per ray: {avg_intensity:.5f}")

Let's also look at the histogram of all ray intensities:

python
plt.hist(intensity, bins=np.linspace(0, 1, 10))
plt.xlabel("Intensity")
plt.ylabel("Number of Rays")
plt.title("Intensity Distribution on Image Surface")
plt.show()
Step
5

Review uncoated results and define the simple-coated doublet class

As we can see, all rays have approximately the same intensity, which is 1. This is because no coatings have been applied to this lens, so no intensity is lost on any surface. This isn't particularly interesting.

Let's now apply a "Simple" coating, which assigns a fixed transmittance and reflectance factor for each surface. Any ray incident on the surface will experience the same transmittance and reflectance. Let's assume the following transmittance values for each surface in our doublet:

  • Surface 1: 50%
  • Surface 2: 60%
  • Surface 3: 80%
  • Surface 4: 90%

We will assume the reflectance is simply 100% minus the transmittance.

python
class DoubletSimpleCoating(optic.Optic):
 def __init__(self):
     super().__init__()

     # define coatings here
     coating1 = coatings.SimpleCoating(transmittance=0.5, reflectance=0.5)
     coating2 = coatings.SimpleCoating(transmittance=0.6, reflectance=0.4)
     coating3 = coatings.SimpleCoating(transmittance=0.8, reflectance=0.2)
     coating4 = coatings.SimpleCoating(transmittance=0.9, reflectance=0.1)

     self.add_surface(index=0, radius=np.inf, thickness=np.inf)

     # add coating 1 to surface 1
     self.add_surface(
         index=1,
         radius=29.32908,
         thickness=0.7,
         material="N-BK7",
         is_stop=True,
         coating=coating1,
     )

     # add coating 2 to surface 2
     self.add_surface(index=2, radius=-20.06842, thickness=0.032, coating=coating2)

     # add coating 3 to surface 3
     self.add_surface(
         index=3,
         radius=-20.08770,
         thickness=0.5780,
         material=("SF2", "schott"),
         coating=coating3,
     )

     # add coating 4 to surface 4
     self.add_surface(
         index=4, radius=-66.54774, thickness=47.3562, coating=coating4
     )
     self.add_surface(index=5)

     self.set_aperture(aperture_type="imageFNO", value=8.0)

     self.set_field_type(field_type="angle")
     self.add_field(y=0.0)
     self.add_field(y=0.7)
     self.add_field(y=1.0)

     self.add_wavelength(value=0.4861)
     self.add_wavelength(value=0.5876, is_primary=True)
     self.add_wavelength(value=0.6563)

     self.image_solve()
6

Define the simple-coated doublet and compare intensity

Let's now define our doublet with "Simple" coatings and perform the exact same analysis.

python
# define the lens0 with coatings
lens1 = DoubletSimpleCoating()

# trace rays through on-axis field point
rays = lens1.trace(Hx=0, Hy=0, wavelength=0.55, num_rays=256, distribution="uniform")

# take intensities of rays after ray tracing
intensity_simple = rays.i

# compute average intensity per ray
avg_intensity_simple = np.mean(intensity_simple)

print(f"Average intensity per ray: {avg_intensity_simple:.5f}")
python
plt.hist(intensity, bins=np.linspace(0, 1, 10), label="No Coating", alpha=0.5)
plt.hist(intensity_simple, bins=np.linspace(0, 1, 10), label="With Coating", alpha=0.5)
plt.xlabel("Intensity")
plt.ylabel("Number of Rays")
plt.title("Intensity Distribution on Image Surface")
plt.legend()
plt.show()
Step
7

Review simple-coating results and define the Fresnel-coated doublet class

The rays propagated through the lens with the coating now have an intensity value of ≈0.216, which is equal to 0.5 * 0.6 * 0.8 * 0.9, as expected. Compare this again to the nominal value of ≈1.0. Note that glass absorption causes a small transmission loss, which is on the order of 1e-4.

Let's define the same doublet, but this time with "Fresnel" coating on each surface. This uses the Fresnel equations to find the transmittance and reflectance coefficients and is equivalent to having uncoating lenses.

There are two key differences with respect to the last coating:

  • We specify coating='fresnel' for each interface surface

  • We must specify the polarization state of the incident light, which we choose as unpolarized light, which means we use a linear combination of two orthogonal polarization states.

    python
    class DoubletFresnelCoating(optic.Optic):
 def __init__(self):
     super().__init__()

     self.add_surface(index=0, radius=np.inf, thickness=np.inf)
     self.add_surface(
         index=1,
         radius=29.32908,
         thickness=0.7,
         material="N-BK7",
         is_stop=True,
         coating="fresnel",
     )  # <-- use fresnel coating
     self.add_surface(index=2, radius=-20.06842, thickness=0.032, coating="fresnel")
     self.add_surface(
         index=3,
         radius=-20.08770,
         thickness=0.5780,
         material=("SF2", "schott"),
         coating="fresnel",
     )
     self.add_surface(
         index=4,
         radius=-66.54774,
         thickness=47.3562,
         coating="fresnel",
     )
     self.add_surface(index=5)
     self.set_aperture(aperture_type="imageFNO", value=8.0)

     self.set_field_type(field_type="angle")
     self.add_field(y=0.0)
     self.add_field(y=0.7)
     self.add_field(y=1.0)

     self.add_wavelength(value=0.4861)
     self.add_wavelength(value=0.5876, is_primary=True)
     self.add_wavelength(value=0.6563)

     self.image_solve()

     # Note the addition of the polarization state here!
     state = PolarizationState(is_polarized=False)
     self.updater.set_polarization(state)
8

Trace rays through the Fresnel-coated doublet

python
lens2 = DoubletFresnelCoating()

# trace rays through on-axis field point
rays = lens2.trace(Hx=0, Hy=0, wavelength=0.55, num_rays=256, distribution="uniform")

# take intensities of rays after ray tracing
intensity_fresnel = rays.i

# compute average intensity per ray
avg_intensity_fresnel = np.mean(intensity_fresnel)

print(f"Average intensity per ray: {avg_intensity_fresnel:.5f}")
9

Plot the Fresnel intensity distribution

As before, let's plot the histogram of the intensity distribution on the image surface.

python
plt.hist(intensity_fresnel, bins=32, label="With Fresnel Coating", alpha=0.5)
plt.xlabel("Intensity")
plt.ylabel("Number of Rays")
plt.title("Intensity Distribution on Image Surface")
plt.xticks(rotation=45)
plt.legend()
plt.show()
Step
10

Plot transmission as a function of aperture position

Clearly, we see a range in transmission through the lens. Let's also look at the transmission as a function of position in the aperture stop.

python
# stop is on first surface --> index 1
x_stop = lens2.surface_group.x[1, :]
y_stop = lens2.surface_group.y[1, :]

plt.scatter(x_stop, y_stop, c=intensity_fresnel, s=5)
plt.axis("equal")
cbar = plt.colorbar()
cbar.set_label("Transmission", rotation=270, labelpad=20)
plt.xlabel("X (mm)")
plt.ylabel("Y (mm)")
plt.show()
Step
Show full code listing
python
import matplotlib.pyplot as plt
import numpy as np

from optiland import coatings, optic
from optiland.rays import PolarizationState

class Doublet(optic.Optic):
  def __init__(self):
      super().__init__()

      self.add_surface(index=0, radius=np.inf, thickness=np.inf)
      self.add_surface(
          index=1,
          radius=29.32908,
          thickness=0.7,
          material="N-BK7",
          is_stop=True,
      )
      self.add_surface(index=2, radius=-20.06842, thickness=0.032)
      self.add_surface(
          index=3,
          radius=-20.08770,
          thickness=0.5780,
          material=("SF2", "schott"),
      )
      self.add_surface(index=4, radius=-66.54774, thickness=47.3562)
      self.add_surface(index=5)

      self.set_aperture(aperture_type="imageFNO", value=8.0)

      self.set_field_type(field_type="angle")
      self.add_field(y=0.0)
      self.add_field(y=0.7)
      self.add_field(y=1.0)

      self.add_wavelength(value=0.4861)
      self.add_wavelength(value=0.5876, is_primary=True)
      self.add_wavelength(value=0.6563)

      self.image_solve()

lens0 = Doublet()
lens0.draw(figsize=(14, 2))

# trace rays through on-axis field point
rays = lens0.trace(Hx=0, Hy=0, wavelength=0.55, num_rays=256, distribution="uniform")

# take intensities of rays after ray tracing
intensity = rays.i

# compute average intensity per ray
avg_intensity = np.mean(intensity)

print(f"Average intensity per ray: {avg_intensity:.5f}")

plt.hist(intensity, bins=np.linspace(0, 1, 10))
plt.xlabel("Intensity")
plt.ylabel("Number of Rays")
plt.title("Intensity Distribution on Image Surface")
plt.show()

class DoubletSimpleCoating(optic.Optic):
  def __init__(self):
      super().__init__()

      # define coatings here
      coating1 = coatings.SimpleCoating(transmittance=0.5, reflectance=0.5)
      coating2 = coatings.SimpleCoating(transmittance=0.6, reflectance=0.4)
      coating3 = coatings.SimpleCoating(transmittance=0.8, reflectance=0.2)
      coating4 = coatings.SimpleCoating(transmittance=0.9, reflectance=0.1)

      self.add_surface(index=0, radius=np.inf, thickness=np.inf)

      # add coating 1 to surface 1
      self.add_surface(
          index=1,
          radius=29.32908,
          thickness=0.7,
          material="N-BK7",
          is_stop=True,
          coating=coating1,
      )

      # add coating 2 to surface 2
      self.add_surface(index=2, radius=-20.06842, thickness=0.032, coating=coating2)

      # add coating 3 to surface 3
      self.add_surface(
          index=3,
          radius=-20.08770,
          thickness=0.5780,
          material=("SF2", "schott"),
          coating=coating3,
      )

      # add coating 4 to surface 4
      self.add_surface(
          index=4, radius=-66.54774, thickness=47.3562, coating=coating4
      )
      self.add_surface(index=5)

      self.set_aperture(aperture_type="imageFNO", value=8.0)

      self.set_field_type(field_type="angle")
      self.add_field(y=0.0)
      self.add_field(y=0.7)
      self.add_field(y=1.0)

      self.add_wavelength(value=0.4861)
      self.add_wavelength(value=0.5876, is_primary=True)
      self.add_wavelength(value=0.6563)

      self.image_solve()

# define the lens0 with coatings
lens1 = DoubletSimpleCoating()

# trace rays through on-axis field point
rays = lens1.trace(Hx=0, Hy=0, wavelength=0.55, num_rays=256, distribution="uniform")

# take intensities of rays after ray tracing
intensity_simple = rays.i

# compute average intensity per ray
avg_intensity_simple = np.mean(intensity_simple)

print(f"Average intensity per ray: {avg_intensity_simple:.5f}")

plt.hist(intensity, bins=np.linspace(0, 1, 10), label="No Coating", alpha=0.5)
plt.hist(intensity_simple, bins=np.linspace(0, 1, 10), label="With Coating", alpha=0.5)
plt.xlabel("Intensity")
plt.ylabel("Number of Rays")
plt.title("Intensity Distribution on Image Surface")
plt.legend()
plt.show()

class DoubletFresnelCoating(optic.Optic):
  def __init__(self):
      super().__init__()

      self.add_surface(index=0, radius=np.inf, thickness=np.inf)
      self.add_surface(
          index=1,
          radius=29.32908,
          thickness=0.7,
          material="N-BK7",
          is_stop=True,
          coating="fresnel",
      )  # <-- use fresnel coating
      self.add_surface(index=2, radius=-20.06842, thickness=0.032, coating="fresnel")
      self.add_surface(
          index=3,
          radius=-20.08770,
          thickness=0.5780,
          material=("SF2", "schott"),
          coating="fresnel",
      )
      self.add_surface(
          index=4,
          radius=-66.54774,
          thickness=47.3562,
          coating="fresnel",
      )
      self.add_surface(index=5)
      self.set_aperture(aperture_type="imageFNO", value=8.0)

      self.set_field_type(field_type="angle")
      self.add_field(y=0.0)
      self.add_field(y=0.7)
      self.add_field(y=1.0)

      self.add_wavelength(value=0.4861)
      self.add_wavelength(value=0.5876, is_primary=True)
      self.add_wavelength(value=0.6563)

      self.image_solve()

      # Note the addition of the polarization state here!
      state = PolarizationState(is_polarized=False)
      self.updater.set_polarization(state)

lens2 = DoubletFresnelCoating()

# trace rays through on-axis field point
rays = lens2.trace(Hx=0, Hy=0, wavelength=0.55, num_rays=256, distribution="uniform")

# take intensities of rays after ray tracing
intensity_fresnel = rays.i

# compute average intensity per ray
avg_intensity_fresnel = np.mean(intensity_fresnel)

print(f"Average intensity per ray: {avg_intensity_fresnel:.5f}")

plt.hist(intensity_fresnel, bins=32, label="With Fresnel Coating", alpha=0.5)
plt.xlabel("Intensity")
plt.ylabel("Number of Rays")
plt.title("Intensity Distribution on Image Surface")
plt.xticks(rotation=45)
plt.legend()
plt.show()

# stop is on first surface --> index 1
x_stop = lens2.surface_group.x[1, :]
y_stop = lens2.surface_group.y[1, :]

plt.scatter(x_stop, y_stop, c=intensity_fresnel, s=5)
plt.axis("equal")
cbar = plt.colorbar()
cbar.set_label("Transmission", rotation=270, labelpad=20)
plt.xlabel("X (mm)")
plt.ylabel("Y (mm)")
plt.show()

Conclusions

First, we see that the uncoated optic has a transmission of near 80%. Secondly, while the effect is small, we can clearly see that at smaller angles of incidence, i.e., near the center, the transmission is a maximum. This is consistent with the Fresnel equations for unpolarized light.

Conclusions:

  • This tutorial introduced the concept of coatings in Optiland.
  • The "SimpleCoating" class was demonstrated, which simply assigns a fixed intensity loss during ray interaction with a surface.
  • The "FresnelCoating" was also introduced, which enables calculation of the effect of e.g., transmission impact due to uncoated glass in lens systems.

The following tutorials will elaborate on the concepts of coatings and polarization in Optiland.

Next tutorials

NextIntroduction to PolarizationTrack the vector state of light as it interacts with birefringent elements.RelatedCustom Optimization OperandsLearn how to optimize your system for specific weight or transmission targets.

Original notebook: Tutorial_6a_Introduction_to_Coatings.ipynb on GitHub · ReadTheDocs