# Lecture 34 - Interference

## Huygen's Principle

For the past few lectures, we studied geometric optics: we have been using the ray model of light, which works well when light interacts with objects that are large compared to the wavelengths being considered.

For objects that have features on the same or smaller length scale as the wavelength of the light there are new effects that cannot be explained by the principles of geometric optics.

When we introduced refraction we invoked Huygen's principle, which explicitly uses a wave model to explain refraction.

Huygen's principle states that every point on a wavefront can be considered to be the centre of a new secondary spherical wave, and the sum of these secondary waves determines the form of the wavefront at any subsequent time.

Click here for a diagram showing how refraction follows from Huygen's principle.

This demo shows how when light enters a medium with a different refractive index, the changing speed results in a changing angle of propagation.

We can also use this model to explain some other important effects that occur, particularly interference and diffraction.

Here is a good demo to get better intuition for Huygen's principle.

One can already see an important aspect of what we will focus on in the next lectures: what happens at the left and right hand side of the sources in the demo above.

## Young's double slit experiment

Thomas Young was an English scientist who provided definitive proof of the wave nature of light by the famous interference experiment, also known as the Young's double-slit experiment.

If we shine monochromatic coherent light on a pair of slits we see an interference pattern.

The key observation can be demonstrated with a laser and two slits, as shown here.

We can treat each of the slits as a point source of circular wavefronts.

A water tank demo nicely demonstrates the concept of two slit interference.

One can understand constructive and destructive interference by working this app.

One can get a feel for what is going on with this app or this app.

Here is a clear demo of how to differentiate between diffraction and interference effects and also how they depend on wavelength.

The condition for constructive interference (bright fringes) is

$d\sin\theta=m\lambda$       (m=0,1,2,..)

and for destructive interference (dark fringes)

$d\sin\theta=(m+\frac{1}{2})\lambda$       (m=0,1,2,..)

## Line spacing for a double slit

For small angles the angles $\theta_{1}$ and $\theta_{2}$ can be approximated as

$\theta_{1}=\frac{\lambda}{2d}$ and $\theta_{2}=\frac{\lambda}{d}$

The positions of the fringes are

$x_{1}=l\tan\theta_{1}\approx l\theta_{1}$ and $x_{2}=l\tan\theta_{2}\approx l\theta_{2}$

So we can see that if the screen is far from the slit the fringes are approximately equally spaced from each other.

## Reflection from a transparent medium

Let us first look at how mechanical waves propagate when they encounter a change in density by looking at this demo.

As in the case of a wave on a rope that is incident on a heavier rope and is reflected with a 180o phase change when a light wave is reflected from a more optically dense media a 180o phase change occurs.

This effect is important when we want to consider interference effects in thin films.

In the absence of interference, the degree to which reflection occurs is given by the reflection coefficient $R$ which at normal incidence for light going from a medium with refractive index $n_{0}$ to one with refractive index $n_{1}$ is given by

$R=(\frac{n_{0}-n_{1}}{n_{0}+n_{1}})^2$

The transmission $T$ and reflectance $R$ add to 1 and represent the fraction of the incident light intensity that is either transmitted or reflected.

## Air wedge and Other Effects

Interference effects can also be observed when light is reflected from the gap between two glass surfaces, which leads to the phenomena known as Newton's Rings.

A similar problem is that of the air wedge, such as shown here.

For a single wavelength dark stripes will occur whenever

$2t=m\lambda$      (m=0,1,2,..)

and bright stripes will occur whenever

$2t=(m+\frac{1}{2})\lambda$      (m=0,1,2,..)

For white light different colors will experience constructive interference at different thicknesses, leading to the colorful lines we see when an air gap is under normal light.

This can be dramatically demonstrated with a soap bubble.

## Anti reflective coating

For most lenses we want as much of the incident light to be transmitted as possible.

Suppose we take a glass lens with refractive index $n=1.52$. We can see from the reflectance equation

$R=(\frac{n_{0}-n_{1}}{n_{0}+n_{1}})^2=(\frac{1-1.52}{1+1.52})^2=0.043$

that about 4% of the incident light is reflected.

This percentage can be reduced by the use of an anti reflective coating.

Ideally we would use a coating that produced an equal amount of reflection at both interfaces, but there is no suitable material with the required refractive index, $n=1.26$, so we use magnesium flouride MgF2.

As the two reflections both occur from more optically dense media they both experience a phase change of $\pi$ on reflection which corresponds to advancing the wave by $\frac{\lambda}{2}$ .

To have the light be out of phase we need to light that goes through the coating to have advanced by $\frac{\lambda}{2}$ for destructive interference to occur.

Critically, when destructive interference occurs the light is not lost, but is instead transmitted.

As the wavelength of light in a medium is given by $\lambda=\frac{\lambda_{0}}{n}$ where $n$ is the refractive index of the medium and $\lambda_{0}$ is the wavelength of the light in free space, the thickness of the coating should be $\frac{\lambda}{4n_{2}}$.

In practice the light incident will not all be the same wavelength, so the thickness of the coating is typically chosen to work optimally in the center of the visible band (~550nm).