# Lecture 31 - Refraction and Lenses

## Refractive index

A useful way to describe a material in terms of it's refractive index. The velocity of light in a medium is related to it's velocity in free space by the refractive index $n$ through the equation

$v=\frac{c}{n}$

Refractive indices of materials are typically somewhere between 1 (vacuum) and ~2.5 (diamond, strontium titanate).

Glass will typically have a refractive index of about 1.5 though the exact value depends on the type of glass.

We can recall that the velocity of a wave is given $v=f\lambda$, when light is traveling in a medium, the frequency $f$ does not change, the wavelength $\lambda$ changes according to

$\lambda=\frac{\lambda_{0}}{n}$

## Huygen's Principle

When light enters a medium with a different refractive index as well as changing it's speed it also changes it's angle.

This can be understood based upon the wave nature of light by applying Huygen's principle.

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.

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

Now see what happens when light travels from vacuum into a medium with a refractive index greater than 1 here.

## Snell's Law

Snell's Law was obtained experimentally quite a long time ago.

Earliest documentation is by Ibn Sahl in 984;

later, Willebrord Snellius in 1621.

It can be now be derived in several different ways from the wave nature of light.

We will simply state it

$$n_{1}\sin\theta_{1}=n_{2}\sin\theta_{2}$$

The refractive index of the medium the light is traveling out of is $n_{1}$.

The refractive index of the material the light is traveling in to is $n_{2}$.

Both the angle of incidence $\theta_{1}$ and refraction $\theta_{2}$ are defined relative to the normal to the surface.

A refraction simulator can be found here and a Wolfram demonstration showing Snell's Law in terms of wavefronts can be found here.

## Refraction through a slab

Light is refracted at both the entrance and exit interfaces when it passes through a slab.

We can see that the angle it exits at is the same as the one it enters at.

Note that while traveling through the slab, the light is bent, which means it comes out of the slab at a different point from it where it would have if it had not been bent.

We can see this is true for all angles of incidence.

## Apparent depth

When viewed from above submerged objects appear to be less deep than the really are.

In the approximation that the angles for the two rays are approximately the same, what is the depth at which our brain will tell us the object is? (our brain does not correct for refraction and assumes light does not bend).

We will also assume that all the angles are small so that we can write $\sin\theta\approx\tan\theta\approx\theta$

From geometric construction we can see that

$\frac{x}{d}=\tan\theta_{1}\approx\theta_{1}$

and

$\frac{x}{d'}=\tan\theta_{2}\approx\theta_{2}$

As Snell's law gives us

$\sin\theta_{2}= n_{1}\sin\theta_{1}\to\theta_{2}\approx n_{1}\theta_{1}$

We can see that $d'=\frac{d}{n_{1}}$. If the object is in water ($n$=1.33) then $d'=\frac{3}{4}d$

## Total internal reflection

For light leaving a more optically dense medium and entering a less optically dense one there is a maximum incident angle, the critical angle $\theta_{C}$ above which light is completely reflected.

We refer to as total internal reflection.

The critical angle can be found from the condition that the refracted angle is $90^{o}$

$\sin\theta_{C}=\frac{n_{2}}{n_{1}}\sin 90^{o}=\frac{n_{2}}{n_{1}}$

## Fiber optics

Optical fibers make use of total internal reflection to confine light to a path.

The fiber is composed of different kind of glass which are doped differently so that the inner fiber has a higher refractive index than the outer fiber.

The concept can be effectively demonstrated with a laser and transparent vat of water; see here.

## Dispersion

Dispersion is an effect where the different colors of light are bent differently due to the fact that the refractive index is not necessarily the same for all wavelengths.

A common example of this is the generation of a spectrum from a white light beam using a prism.

Rainbow's are also examples of dispersion and are generated by light reflected within spherical raindrops.

Another atmospheric dispersion phenomena is the green flash which can be seen (rarely) at sunset.

More on green flashes here and an explanation here .

## Thin lenses

A lens makes use of the refractive properties of a material to focus light.

The basic building blocks of lenses are faces that are either spherical or planar.

The spherical faces can either curve outwards (convex) or inwards (concave).

The key property of a lens is that it focuses rays traveling parallel to the axis to a particular point, the focal point.

The focal point is a distance $f$ from the center of the lens and is defined as the focal length of the lens.

A related measurement of the focusing ability of a lens is the power $P$

$P=\frac{1}{f}$

measured in Diopters ($\mathrm{D}$) which are equivalent to $\mathrm{m^{-1}}$

## Converging lens

A lens which is thicker in the center than at the edges is a converging lens, an incoming parallel beam of light will be focused to a point $F$ at $x=f$ from the center of the lens.

To obtain a parallel beam of light light should be radially propagating outwards from the point $F'$ a distance $x=-f$ from the center of the lens.

Converging Lens applet.

## Diverging Lens

A lens which is thinner in the center than at the edges is a diverging lens, an incoming parallel beam of light will be focused to a point $F$ at $x=-f$ from the center of the lens, which means that the rays appear to diverge outwards from that point.

To obtain a parallel beam of light light the incoming rays should have a path such that they would go to the point $F'$ a distance $x=f$ from the center of the lens if the lens were not there..

Diverging Lens applet.