Waves Tutorial 6 – Reflection, Refraction, and Optical Fibres

We have seen in Unit 1 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 × 10⁸ m/s in air.  When light hits an object it is:

• Reflected

• Transmitted

•  Absorbed.

Reflection

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, q₁ = q₂

• 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!

Refraction

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 × 10⁸ m/s, while in glass its speed is 2 × 10⁸ m/s.  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.

Snell’s Law

It was the Dutch physicist Snell, who in 1621 established the quantitative relationship between the angle of incidence and the angle of refraction.  If we double the angle of incidence, we do NOT double the angle of refraction

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 × 10⁸ m/s ÷ 1.5 = 2 × 10⁸ m/s.

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:

                                    n₁ = c/c₁

and          

                                    n₂ = c/c₂

  We can rearrange each equation and combine the two to write:

n₁c₁ = c = n₂c₂

  So we can write:

    n₂ = c₁

    n₁    c₂

The ratio n₂/n₁ is the relative refractive index, which we can also define as the ratio of the sines of the angles of incidence and refraction:

n₂ = c₁  = sinq₁

n₁    c₂    sinq₂

We can write this formula in a more useful form:

n₁ sin q₁ = n₂ sin q₂

The ratio n₂/n₁ is the relative refractive index, which we can also write as 1n2. This means the refractive index of light going in from material 1 to 2.

₁n₂  = n₂ = c₁  = sin q₁

          n₁    c₂       sin q₂

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:

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:

 ₁n₂ = sin q₁

          sin q₂

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.

  The relative refractive index going from medium 2 to medium 1 is ₂n₁, so we can write:

  ₂n₁ = sin q₂

           sin q₁

 Therefore:

 ₁n₂ = 1 ÷ ₂n₁ 

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

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:

n₁ sin q₁ = n₂  sin q₂

At the critical angle:

n₂ sin q_c = n₁  sin 90

Since sin 90 = 1, we write:

n₂ sin q_c = n₁

Þ sin q_c = n₁

                 n₂

We can write this as:

sin q_c =   1

                       n₂/n₁

Since n₂/n₁ is the relative refractive index ₁n₂, we can write

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 infra-red transmitters.

• The light is channelled along a very thin central fibre that is cladded 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 fly-by-light 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:

n =    1

           sin q_c

Where n is the physics code for the refractive index.

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.

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.

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.