Thin-films are a wonder of nanotechnologies: discover the properties of an anti-reflective coating with our thin-film optical coating calculator.

Here you will learn everything you need about interference in thin-films. Keep reading to find out:

  • What happens at the interfaces of a system of three materials with different refractive indexes;
  • What is interference, and why is interference in thin-films is so useful;
  • How to use the thin-film interference equations to predict the behavior of a material;
  • A step-by-step guide to our thin-film optical coating calculator.

And much more!

The (optical) path to interference in thin-films: anti-reflective nanostructures

Let's define our physical set-up first. We have a structure of three materials, each with a different refractive index. We can assume a situation in which:

  • The first material is air, with refractive index n1n_1;
  • The second material is a thin film with refractive index n2n_2 and thickness dd; and
  • The third material is a "substrate" with refractive index n3n_3.

We identify two interfaces:

  • The n1n2n_1-n_2 interface; and
  • The n2n3n_2-n_3 interface.

If a light beam with wavelength λ\lambda hits the material with an angle of incidence θ\theta, at both interfaces, we see a partial transmission and reflection. At this point, the different refractive indexes of the materials become relevant since they modify the optical path of the beam.

We can define the difference in phase between the first and the second reflected beams with the formula for the optical path difference (OPD):

OPD=2n2dsin(θ2)\text{OPD} = 2\cdot n_2\cdot d\cdot \sin{(\theta_2)}

Where θ2\theta_2 is the angle of the refracted beam in the second material that you can calculate with the Snell's law.

Now we have two reflected light beams traveling outward, one of them after passing back and forth in the second material, with an added distance equal to the optical path difference (this difference arises because the velocity of light changes, introducing a displacement). Since light is a wave (but also a particle, ask Louis de Broglie), it is characterized by a phase. The difference in path introduced by the refraction-reflection also implies a phase shift. Now the two reflected beams can interfere with each other.

How to calculate the phase change in thin-films interference?

Depending on the structure you are analyzing, you can find various situations. In the table below, you can see how combinations of the refractive indexes of the three materials affect the behavior of reflected light.

180°180\degree Phase shift

At top and bottom interfaces

At top or bottom interface

Refractive indices

n1<n2,n2<n3n_1\lt n_2, n_2\lt n_3

n1>n2,n2<n3n_1\gt n_2, n_2\lt n_3

n1<n2,n2>n3n_1\lt n_2, n_2\gt n_3

OPD for constructive interference

mλm\lambda

(m12)λ\left(m-\frac{1}{2}\right)\lambda

OPD for destructive interference

(m12)λ\left(m-\frac{1}{2}\right)\lambda

mλm\lambda

mm is any positive integer: constructive and destructive interference happens for any thickness of the thin film that allows for the same phase difference to arise.

How to use the equations for interference in thin-films to predict the behavior of a material

Let's talk interference first; then, we will check the possible outcomes of our thin-film set-up.

Interference is a physical phenomenon in which two (or more) waves reinforce or damp each other according to their relative difference of phase. We can identify two extremes in the interference behavior:

  • Constructive interference, where the two waves are in phase and when they meet, their amplitudes sum; and
  • Destructive interference, where the waves are completely out of phase and cancel each other when they meet.

In between the two situations, we find an intermediate regime.

In our set-up, we need to add another factor in our interference analysis: if the second material in an interface has a lower refractive index than the first one, then the light undergoes a 180°180\degree or π2\tfrac{\pi}{2} phase shift.

How to use thin-film interference: the anti-reflective coating thickness

Interference in thin films has extensive use in producing anti-reflective coating. To do so, the thin film is engineered in such a way as to obtain destructive interference over the widest possible range of wavelengths.

For a specific wavelength λ\lambda, assuming a set-up where n1<n2<n3n_1<n_2<n_3, we know that the thickness for destructive interference depends on the relation:

2n2dcos(θ2)=(m12)λ2\cdot n_2\cdot d\cdot\cos{(\theta_2)}=\left(m-\frac{1}{2}\right)\lambda

Substituting m=1m=1, we obtain the minimal thickness of the layer with the formula:

dmin=λ4n2d_{\text{min}} = \frac{\lambda}{4\cdot n_2}

Reflectivity and transmissivity

We can calculate the percentages of the total intensity of light transmitted or reflected by the medium at an interface. For s-polarized and p-polarized light, we have, for an 121-2 interface, the following relations for the reflectivity:

Rs=n1cosθ1n2cosθ2n1cosθ1+n2cosθ22Rp=n1cosθ2n2cosθ1n1cosθ2+n2cosθ12\quad\begin{align*} R_s = \left|\frac{n_1\cos\theta_1-n_2\cos\theta_2}{n_1\cos\theta_1+n_2\cos\theta_2}\right|^2\\ R_p = \left|\frac{n_1\cos\theta_2-n_2\cos\theta_1}{n_1\cos\theta_2+n_2\cos\theta_1}\right|^2 \end{align*}

And the following ones for the transmissivity:

Ts=1RsTp=1Rp\quad\begin{align*} T_s = 1-R_s\\ T_p = 1-R_p \end{align*}

In a three mediums set-up, the total reflectivity depends on both the fraction reflected by the first interface and the one reflected by the second one:

Rs=Rs1+Rs2Rp=Rp1+Rp2 \begin{align*} R_s &= R_{s1} + R_{s2}\\ R_p &= R_{p1} + R_{p2} \end{align*}

Where the numbers in subscript mark the interface interested by the reflection.

How to use our thin-film optical coating calculator

With our thin-film optical coating calculator, you can easily calculate the interference in thin-films: insert the values you know: usually wavelength, thin-film thickness, angle of incidence, and refractive indexes. Our calculator will tell you the type of interference in the thin-film structure (constructive, destructive, or intermediate) and suggest the minimum thickness to achieve anti-reflection.

Davide Borchia
Incident Angle (θ₁)
deg
Optical film thickness (d)
nm
Wavelength and refractive indices
Wavelength (λ)
nm
Refractive index of first medium (n₁)
Refractive index of optical film (n₂)
Refractive index of substrate (n₃)
OPD and reflectivity
Optical path difference (OPD)
nm
s-polarized reflectivity (Rₛ)
%
p-polarized reflectivity (Rₚ)
%
Interference type
Min anti-reflection coating thickness
nm
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