Anti-Reflective Coating with Multiple Layers

An anti-reflective coating is a set of thin, transparent films applied to the surface of an optical device such as a lens to reduce reflection. This reduction of reflected light leads to an increase in the efficiency of the optical system and minimizes stray light, which is important in many imaging applications. Anti-reflective coatings can also be applied to the surfaces of eyeglasses to reduce glare and make the eyes of the wearer more visible.

The simplest example of an anti-reflective coating is a quarter-wavelength layer, a single dielectric film with thickness equal to one quarter of the wavelength of the incident light. This layer can reduce the reflection coefficient to zero if the refractive index of the film is equal to the geometric mean of the refractive indices of the air (n0) and substrate (nS). For air (1.0) and common glass substrate (1.5) this optimal refractive index would be sqrt((1.0)(1.5)) or approximately 1.22.

Typically no material exists with a refractive index that yields a reflection coefficient of exactly zero. Another drawback of the quarter-wavelength layer is that while it can prevent reflection of light at one wavelength, it reflects a substantial amount of radiation at any other nearby wavelength. The reflectance of the quarter-wavelength layer also depends heavily on the angle of incidence of the light.

An alternative is to use a coating that consists of multiple layers. Compared to single-layer coatings, a multilayer coating is more likely to reduce the reflection coefficient across a band of wavelengths and can be produced using a wider variety of real materials.

In this tutorial, the reflectance of two different multilayer coatings is compared over a wide spectral range: a quarter-quarter coating (two layers), and a quarter-half-quarter coating (three layers). The quarter-half-quarter coating is shown to have more consistently low reflectance across most of the visible spectrum.

Model Definition

Figure 1 shows the simple geometry used in this model. It consists of a box with an interior boundary separating the air and substrate domains. This boundary is also where the features are added.

The red arrow (activated in the feature) shows the sense in which the thin dielectric layers are stacked. The last Thin Dielectric Film in the Model Builder represents the topmost thin film in the model.

Figure 1: Geometry used to model multilayer anti-reflective coatings.

Simple anti-reflective coatings are designed to minimize the reflectance at a specified vacuum wavelength, λ0. More sophisticated coatings can minimize reflectance across a relatively large band of wavelengths by employing multiple thin films with different material properties.

In this tutorial, two different types of multilayer antireflective coating are investigated. The first is a quarter-quarter coating with two layers of different refractive index on the surface of a glass substrate. The name “quarter-quarter coating” comes from the fact that each layer has a thickness of λ0/(4n), where n is the refractive index of the layer.

Theoretically, a quarter-quarter coating can reduce the reflectance to zero at the specified wavelength; however, because real materials with ideal refractive indices are seldom available, zero reflectance is normally not achieved.

In this example, the substrate consists of glass (ns = 1.5) and the first thin dielectric layer is made of magnesium fluoride, MgF2 (n1 = 1.38). The following expression can be used to determine the optimal refractive index for the second layer, n2, so that the reflectance can be reduced to zero:

where the refractive index of the air, na, is taken to be 1.

Using this expression the optimal value for n2 is 1.69. From Table 1, cerium fluoride, CeF3, has a refractive index close to this value, 1.63. Using this material a minimum reflectance of less that 1% can be achieved.

Table 1: Refractive indices of Materials Frequently used in Thin Films.
Material	Refractive index
Magnesium Flouride (MgF2)	1.38
Silicon Dioxide (SiO2)	1.46
Cerium Fluoride (CeF3)	1.63
Zirconium Oxide (ZrO2)	2.2
Silicon (Si)	3.5

The reflectance as a function of vacuum wavelength is shown in Figure 2. A noticeable drawback of the two-layer coating is that the reflectance is only significantly reduced in a narrow band around a single wavelength.

Figure 2: Reflectance of a quarter-quarter coating.

The reflectance can be reduced over a wider range of wavelengths by using a dielectric film with three or more layers. An example of a three-layer coating is the quarter-half-quarter coating, in which a thin layer of thickness λ/2 is placed between the two quarter-wavelength layers.

Results and Discussion

Figure 3 compares the reflectance of the quarter-quarter and quarter-half-quarter films. Because the refractive indices of real materials are used, the reflectance of the quarter-quarter coating does not decrease to zero. The quarter-half-quarter film exhibits slightly greater reflectance at the center of the band, but the reflectance remains lower than 1% over a much wider spectral range.

Figure 3: Reflectance response for a quarter-quarter and quarter-half-quarter coating configurations.

Ray_Optics_Module/Prisms_and_Coatings/antireflective_coating_multilayer

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
n_air	1	1	Refractive index of air
n_glass	1.5	1.5	Refractive index of glass
n_CeF3	1.63	1.63	Refractive index of CeF3
n_MgF2	1.38	1.38	Refractive index of MgF2
n_ZrO2	2.2	2.2	Refractive index of ZrO2
lam0	550[nm]	5.5E-7 m	Vacuum wavelength

Geometry 1

Square 1 (sq1)

In the toolbar, click

In the window for , click to expand the section.

In the table, enter the following settings:


Layer name	Thickness (m)
Layer 1	0.5

Click

Materials

Material 1 (mat1)

In the window, under right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Refractive index, real part	n_iso ; nii = n_iso, nij = 0	n_air	1	Refractive index

Material 2 (mat2)

Right-click and choose .

Select Domain 1 only.

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Refractive index, real part	n_iso ; nii = n_iso, nij = 0	n_glass	1	Refractive index

Geometrical Optics (gop)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

In the text field, type 0.

Set up the quarter-quarter anti-reflective coating using two features.

It is useful to display the boundary normal in the window since this indicates the order in which the thin films are arranged. The arrow points in the direction away from the substrate, so the first layer that appears in the Model Builder is adjacent to the substrate and the last layer that appears is adjacent to the air domain.

Material Discontinuity 1

In the window, under click .

In the window for , locate the section.

Select the check box.

Locate the section. From the list, choose .

Locate the section. From the list, choose . It is still possible to analyze the reflectance of the coating based on the intensity of the transmitted rays, even if no reflected rays are actually released at the boundary.

Thin Dielectric Film 1

In the toolbar, click

and choose .

The first layer (directly touching the glass substrate) is the CeF3 layer with a refractive index of 1.63. The thickness is set to be a quarter of the wavelength in this material.

In the window for , locate the section.

In the n text field, type n_CeF3.

In the t text field, type lam0/(4*n_CeF3).

Material Discontinuity 1

In the window, click .

Thin Dielectric Film 2

In the toolbar, click

and choose .

The second layer (on top of layer 1) is the MgF2 layer with a refractive index of 1.38. The thickness of this layer is also set to a quarter of the wavelength.

In the window for , locate the section.

In the n text field, type n_MgF2.

In the t text field, type lam0/(4*n_MgF2).

Next, set up the feature to release a number of rays of different wavelengths from 400 nm to 800 nm.

Release from Grid 1

In the toolbar, click

and choose .

In the window for , locate the section.

In the qx,0 text field, type 0.5.

In the qy,0 text field, type 1.

Locate the section. Specify the L0 vector as


0	x
-1	y

Locate the section. From the list, choose .

Click

In the dialog box, choose from the list.

In the text field, type 400[nm].

In the text field, type 800[nm].

In the text field, type 100.

Click .

Set up a study step to compute the ray trajectories to a maximum optical path length of 1.1 m.

Study 1

Step 1: Ray Tracing

In the window, under click .

In the window for , locate the section.

From the list, choose .

In the text field, type range(0,0.01,1.1).

In the toolbar, click

Results

Reflectance

In the toolbar, click

and choose .

In the window for , type Reflectance in the text field.

Locate the section. From the list, choose .

From the list, choose .

Click to expand the section. From the list, choose .

In the text area, type Reflectance of Multilayer Films.

Locate the section. Select the check box.

In the associated text field, type Vacuum wavelength (nm).

Select the check box.

In the associated text field, type Reflectance (%).

Ray 1

In the toolbar, click

and choose .

Plot the percentage reflectance.

In the window for , locate the section.

In the text field, type 100*(gop.relg1.Q0-gop.Q)/gop.relg1.Q0.

Locate the section. From the list, choose .

In the text field, type gop.lambda0.

From the list, choose .

Click to expand the section. Select the check box.

From the list, choose .

In the table, enter the following settings:


Legends
Quarter-Quarter

In the toolbar, click

. The plot should look like Figure 2.

Next, to simulate a quarter-half-quarter layer, add another feature. The material of the middle layer is chosen to be Zirconium Oxide, ZrO2, with refractive index 2.2. Set the thickness to half the specified wavelength.

Geometrical Optics (gop)

Material Discontinuity 1

In the window, under click .

Thin Dielectric Film 3

In the toolbar, click