Contents Total Internal Reflection in Optical Fibres Dispersion 
We have seen in Particle Physics how light behaves as a particle and how particles behave as waves. In this tutorial, we are going to look at how light shows wave behaviour. All other waves behave in the same way.
Some Revision
In this section we will consider light to be a wave carrying energy that travels in straight lines at a speed of 3.0 × 10^{8} m s^{1} in air. When light hits an object it is:
Reflected
Transmitted
Absorbed.
You
should be familiar with reflection which you did in early secondary school.
All reflection depends on the smoothness of the surface.
If the surface is smooth, then parallel rays are reflected parallel (specular
reflection). If the surface is
rough, then parallel rays are scattered.
However each individual ray obeys the law
of reflection.
High quality mirrors are silvered on the front to prevent multiple images. Here is a ray of light striking a mirror at an angle:
We observe the following:
The
angle of incidence = angle of reflection,
q_{1}
= q_{2}
The
reflected ray is in the same plane
as the incident ray.
All
angles are measured from the normal,
a line running at 90^{o} to the surface.
The
image in a mirror has these features:
It
is the same distance behind the mirror line as the object is in front.
It
is the same size as the object.
It
is the right way up (erect).
It
is laterally inverted, i.e. left and right swapped over.
It
is virtual, meaning that the rays cannot be projected onto a screen.
The image is not really behind the mirror; it just appears
to be.
Curved mirrors obey the same rules of reflection as flat (plane) mirrors do.
Reflections can give a lot of atmosphere to a photograph  what an excuse to share one of my favourites!
These notes in this section are for students doing the Irish syllabus.
Before we look at the reflecting telescope, we should have a quick revision at curved mirrors. There are two kinds:
The concave mirror. You may well have one in the shaving mirror in the bathroom;
The convex mirror. You will see these used as security mirrors in shops.
A concave mirror brings parallel rays of light together.
Each ray obeys the Law of Reflection. You can see that the rays come together, or converge.
Note how the shape of the mirror brings all the rays to a single point called the principal focus. The distance between the principal focus and the surface of the mirror is called the focal length.
If the object is close up to the mirror, it appears the right way up (upright or erect) and is magnified (made bigger). If it's further away the image is upside down (inverted) and diminished (made smaller).
Other waves can be reflected by a concave mirror. A satellite dish is a concave mirror to reflect microwave waves onto an antenna. There was a device produced after the First World War to focus sound waves of incoming aircraft to give early warning of their presence. And it worked.
Convex Mirror
A convex mirror reflects light rays outwards as shown in the diagram.
If we extend the rays behind the mirror, we see that they meet at a principal focus. The image is virtual, upright, and diminished.
We can draw the ray diagrams with accurate drawing, but we need to know the shape of the mirror. It is easiest if the mirror is considered to be spherical, rather than parabolic. The focal length of a spherical mirror is half the radius.
You will, of course, need a compass to draw the mirror line on your graph paper. Remember that the angle of incidence with the normal line to the surface of the curved mirror is the same as the angle of reflection from the normal line. (Drawing the normal line to the curved surface is easier said than done.)
The curved mirror formula is of similar form to the lens formula:
[f  focal length (m); u  object distance (m); v  image distance (m)]
The sign conventions are:
f is positive for a concave mirror;
f is negative for a convex mirror;
v is positive for a real image;
v is negative for a virtual image.
The magnification (ratio of image height to object height) is given by:
Worked example A concave mirror is spherical with a radius of 30 cm. A light bulb is 3.0 cm high and is placed 12.0 cm from the mirror on the principal axis. What is the image distance? What is magnification? What is the image height? Is the image real or virtual? 
Answer Focal length needs to be worked out: f = 30 cm ÷ 2 = 15 cm Since it's a concave mirror, the focal length is positive.
Now we use the equation:
(15 cm)^{1} = (12 cm)^{1} + v^{1} v^{1} = 0.0667 cm^{1}  0.0833 cm^{1} = 0.0167 cm^{1} v = 60 cm
Magnification Equation:
M = 60 cm ÷ 12 cm = 5.0
Image height = 3.0 cm × 5.0 = 15 cm (ignoring the minus sign)
The image is virtual because the sign for the image distance is negative. It is the right way up.
(Strictly speaking the magnification equation is: M =  v/u =  60 cm/ 12 cm = +5.0)

If we place the object on the other side of the principal focus (i.e. u > f) the image is real, magnified, and inverted.
Now we will do the same for a convex mirror.
Worked example A convex mirror is spherical with a radius of 30 cm. A light bulb is 3.0 cm high and is placed 12.0 cm from the mirror on the principal axis. What is the image distance? What is magnification? What is the image height? Is the image real or virtual? 
Answer Focal length needs to be worked out: f = 30 cm ÷ 2 = 15 cm Since it's a convex mirror, the focal length is negative.
Now we use the equation:
(15 cm)^{1} = (12 cm)^{1} + v^{1} v^{1} = 0.0667 cm^{1}  0.0833 cm^{1} = 0.15 cm^{1} v = 6.67 cm
Magnification Equation:
M = 6.67 cm ÷ 12 cm = 0.556
Image height = 3.0 cm × 0.556 = 1.67 cm (ignoring the minus sign)
The image is virtual because the sign for the image distance is negative. It is the right way up.
(Strictly speaking the magnification equation is M =  v/u =  6.67 cm/ 12 cm = +0.556)

Light rays are bent when they travel from a medium of one optical density into another, for example from air to glass. In air light travels at 3.0 × 10^{8} m s^{1}, while in glass its speed is 2.0 × 10^{8} m s^{1}. We say that glass in an optically denser medium than air. We can see that the direction of the ray is abruptly changed, or deviated as it passes the boundary:
We should note the following:
The
ray bends in towards the normal as the ray enters the glass.
The angle of incidence is greater than the angle of refraction.
On
leaving the glass the light regains its original speed.
The emergent ray bends
away from the normal. In
this case the angle of incidence is less than the angle of refraction.
The
path of the emergent ray is parallel to the path of the undeviated ray
(where the ray would have gone if there hadn’t been a piece of glass in
the way). This only happens when the sides of the object are parallel.
It wouldn’t happen in a prism.
If the
boundary is indistinct, then refraction is gradual with the change occurring
over a long distance. Earthquake
waves refract in curves due to the gradually changing density of rocks.
Different wavelengths of light are refracted by different amounts; blue light is refracted the most, red light the least.
When doing calculations we generally assume the light is yellow.
There is always a weak reflected ray.
Can you complete these diagrams showing the refracted ray and the emergent ray?

Refractive Index
The refractive index given the physics code n and has no units. We can also describe the absolute refractive index as the ratio of the speed of light in a vacuum to the speed of light in an optical medium. Glass has a refractive index of 1.5, so the speed of light is:
3.0 × 10^{8}
m s^{1} ÷ 1.5 = 2.0 × 10^{8} m s^{1}.
Before we carry on, we need to be sure of what these terms mean:
The
absolute refractive index is the ratio compared with the refractive
index of a vacuum. (n
for a vacuum = 1.000;)
The
relative refractive index is the ratio of the absolute refractive
index of one material compared to that of another, for example from water to
glass.
From this we can write:
and
We can rearrange each equation and combine the
two to write:
Note that:
Absolute refractive index is the ratio between the speed of light in a vacuum and speed of light in a material.
It is always greater than 1.0, as there is no material optically less dense than a vacuum.
The relative refractive index is the ratio between the speed of light in material 1 and the speed of light in material 2.
Relative refractive index can be less than 1.
There are no units for refractive index as it's m s^{1} / m
s^{1}.
What is the speed of light in glass, refractive index 1.5, when the speed of light in air is 3.0 x 10^{8 } m s^{1}? 
Snell’s
Law
It was the Dutch physicist Willebrord Snel van Royen (1580 – 1626), who in 1621 rediscovered the quantitative relationship between the angle of incidence and the angle of refraction. It was initially worked out by by Ibn Sahl (940 – 1000), an Arabic scholar, in 984. If we double the angle of incidence, we do NOT double the angle of refraction.
The
ratio
n_{2}/n_{1} is
the relative refractive index, which we can also define as the ratio of
the sines of the angles of incidence and refraction:
We can write this formula in a more useful form:
n_{1} sin q_{1} = n_{2} sin q_{2}
The ratio n_{2}/n_{1} can also be written as as _{ 1}n_{2}. This means the refractive index of light going in from material 1 to 2.
What does _{2}n_{1} mean? 

A ray of light strikes an airglass boundary at an incident angle of 30^{o}. If the refractive index of glass is 1.5, what is the angle of refraction? 
Remember
that each wavelength of light has a slightly different refractive index.
Generally we use yellow light.
The
table shows some absolute refractive indices for some common materials:
Medium 
Refractive
Index 
Vacuum 
1.000 
Air 
1.003 
Ice 
1.30 
Water 
1.33 
Glass 
1.50 
Diamond 
2.42 
The absolute refractive index will always be always be greater than 1.0 as light cannot travel any faster then it does in a vacuum. It is possible for particles to travel faster than light in a material (but NOT in a vacuum). The result of this will be little flashes, the light equivalent of a sonic boom, called Cherenkov radiation. This is why radioactive materials appear to make water glow.
Photo: United States Nuclear Regulatory Commission, Wikimedia Commons
Now let us have a look at what happens when a ray passes from a dense medium to a less dense medium. Diagram A shows the ray going from air to glass, while Diagram B shows the ray leaving the glass and going into air.
In Diagram A we have
established that:
In Diagram B we see:
Ray
of light travels faster in the optically less dense medium
The
ray bends away from the normal.
The
angle of incidence is less than
the angle of refraction.
Therefore:
_{1}n_{2} = 1 ÷ _{2}n_{1}
For example, the refractive index from air to water is 1.33, while the refractive index from water to air = 1/1.33 = 0.75
What is the relative refractive index when a light ray passes from glass (RI = 1.5) to air? 

What is the angle of refraction and what is the angle through which the ray is deviated when light passes at an incident angle of 48^{o} into water of refractive index 1.33 from air at a refractive index of 1.00? 
Critical Angle
We have seen how a ray of light passing from glass to air bends away from the normal. If we increase the angle of incidence, the angle of refraction increases more (Diagrams A and B):
Angle of refraction getting bigger:
At
a particular value of angle of incidence, the angle of refraction is 90^{o},
as shown in Diagram C. This
particular angle of incidence is called the critical
angle.
Above the critical angle we get total internal reflection. There is no transmission of light at all. See diagram D.
Above the critical angle we
get total internal reflection. There
is no transmission of light at all. We know that:
n_{1}
sin
q_{1}
= n_{2 } sin
q_{2}
At the critical angle:
n_{2}
sin
q_{c}
= n_{1 } sin 90
Since sin 90 = 1, we write:
n_{2}
sin
q_{c}
= n_{1}
Therefore
We can write this as:
Since n_{2}/n_{1} is the relative refractive index _{1}n_{2}, we can write:
Or we can write:
If you struggle with this, the best way to tackle it is to use:
n_{2}
sin
q_{c}
= n_{1 } sin 90
Note:
Make sure that the bigger refractive index goes downstairs in the equation, otherwise you will get a sine value greater than 1. This will not work.
So if you get sin^{1} (1.06), for
example, you have done it wrong.
What is the critical angle of water, of which the refractive index is 1.33? 

Rays from a point source of light at the bottom of a swimming pool 1.8 m deep strike the water surface and only emerge through a circle of radius r, as shown in the diagram. If the refractive index between water and air is 1.33, calculate: (a) the critical angle, q_{c}; (b) the value of r.
Give your answers to an appropriate number of significant figures. 
Total Internal Reflection in Optical Fibres
An example of the use of total internal reflection is in optical fibres. At their simplest, optical fibres are long thin strands of glass that carry light from one end a long distance to the other. The light can be guided round corners using total internal reflection.
There are two problems:
Ordinary glass is not very good. It is impure and has a high attenuation coefficient, losing most of its intensity after a short distance. Red and blue light is absorbed, leaving only green.
Wide
fibres tend to smear the signals as there is a path difference between the
rays that go down the
middle of the fibre and those that bounce from side to side.
The problems can be overcome by:
Using
very high purity glass.
Using
infrared transmitters.
The
light is channelled along a very thin central fibre that is clad with
glass of a lower refractive index. Fibres
with a step index have one layer
of cladding, while graded index
fibres have several layers of cladding.
The best fibres are called monomode
fibres, with a channel of no more than 1.25
mm, which is very narrow.
The path difference between the axial ray and the reflected rays is
negligible.
Optical fibres are amazingly flexible and strong. They mounted in a polythene tube for further strength. Optical fibres offer many advantages over wires.
Fly by Light
If we use a copper wire to transmit data from several sensors, we have to be very careful to screen it from unwanted signals. Ordinary wires are very prone to this, especially in long runs. Screening is easy enough; we simply surround the wire with an insulating layer and then copper braiding to shield the wire. We have a coaxial cable.
The problem with coaxial cable is that:
It is heavy;
It is bulky. Lots of coaxial cables take up a lot of space.
More recently there have been projects that use flybylight technology. This uses optical fibres, which are much less heavy and take up a lot less space; several optical fibres can take the space of one coaxial cable. An optical fibre looks like this:
The fibres are 50 mm across (including the sheath). The core has a diameter of 5 mm (5 × 10^{6} m).
Optical Fibres
Optical fibres work by total internal reflection. The light ray makes a certain angle of incidence when it hits the boundary of an optically dense material (like glass) and an optically less dense material (like air). If this angle is greater than the critical angle, the ray is totally internally reflected. The critical angle, q_{c}, is determined by the formula:
Where _{1}n_{2} is the physics code for the refractive index going from material 1 to material 2. We can rewrite the equation in a more userfriendly way:
Or we could go back to our old friend:
n_{1} sin q_{1} = n_{2} sin q_{2}
The rays of light should travel like this:
But instead light rays can travel several paths:
This means that the light rays can arrive at different times, resulting in dispersion or smearing. The signal that was sharp when it left the transmitter is smeared. It is called modal dispersion.
The picture shows us how the signal can be unacceptably distorted and even produce spurious signals that were not there. The problem can be resolved by cladding the core with a material of slightly lower refractive index. For example the core might have a refractive index of 1.6, while the cladding has a refractive index of 1.4. This is called a step index fibre.
Dispersion can be reduced further by use of a graded index or multimode fibre. Some light is passes down the middle, which has a higher refractive index, therefore slower rate of travel. With clever manipulation of the refractive indices, the ray travelling down the middle can be made to arrive at the same time as the ray that goes from side to side. They can meet with a time difference of about 1 ns km^{1}. In aircraft where the distances are generally less than 50 metres, this is not too bad.
Monomode fibres are designed such that the rays pass only down the middle. If the light were perfectly monochromatic, i.e. of one wavelength only, the rays would all arrive at the same time. However even the best lasers produce a slight spread, and since refractive index varies with wavelength, there can be slight differences in arrival times, leading to smearing. This is called chromatic dispersion.
Material dispersion happens when different wavelengths interact with the material of the optical fibre in slightly different ways. This can lead to smearing of the signal.
Worked Example The picture shows light entering into a straight length of stepindex optical fibre.
The critical angle between the core, refractive index 1.52, and the cladding is 59^{o}. (a) Calculate the refractive index of the cladding. (b) What happens to a light ray that strikes the core/cladding boundary at an angle of less than 59^{o}? (c) What happens to a light ray that strikes the core/cladding boundary at an angle of greater than 59^{o}? 
Answer (a) Use our old friend: n_{1} sin q_{1} = n_{2 }sin q_{2}
n_{1} sin 90 = 1.52 sin 59
n_{1} = 1.52 × 0.857 = 1.30 (b) The light ray will be refracted into the outer cladding. (c) The light ray will be totally internally reflected. 
A stepped index optical fibre has a central core of refractive index 1.50, and a cladding of refractive index 1.45.
A single fine beam of monochromatic light enters the core at an incident angle of 30^{o}. (a) Calculate the angle of refraction of the light ray as it passes into the fibre. (b) Calculate the critical angle between the core and the cladding. (c) Explain whether or not the light will continue to go down the core. 
Although we have looked at light, all these phenomena can be observed with other forms of waves, e.g.
radio waves;
water waves;
sound waves.