Refraction
Have you ever seen "broken" spoon in a glass? Or maybe you have tried
to catch fish with your hands and you could not. Why it is like that?
This illusion is produced by phenomenon called refraction of light. Refraction
is bending of a light beam as it passes from one material into another.
Generally: when light passes from a medium (material) of higher density to the one of lower
it bends off the normal (the perpendicular to the surface it strikes) and when it passes from medium
of lower density into the one of higher it bends to the normal.
The incoming ray is called incident ray and the angle it makes with the normal is
called the angle of incidence. Ray after bending is called refracted ray and the angle
it makes with the normal is called the angle of refraction. Imortant thing is that the
incident ray, the refracted ray and the normal to the surface are all in
the same plane (this sentence is called the first law of refraction). Interesting thing is that if angle of
incidence is 0 degree (incendent ray is parallel to the normal), it is not refracted, it does not change
its direction.
The spoon in this glass seems to be broken.
Now you can try this experiment. Take some open container (the best would be
a pot) and put a coin in the middle of the bottom and look at it. Now move
your head as long as you can see the coin. When it disappears pour the water in the pot very slowly and
remember not to move your head. When there is enough water in the pot, you can
see the coin again! But coin appears to be bigger and nearer than it is in
reality. This effect is called floating-coin illusion. This happens because rays reflected
from the coin are refracted as they leave water and you can see them.
Floating-coin illusion.
Many people have tried to find geometrical connection between the angle of incidence
and the angle of refraction. First one was Ptolemy, but his law of refraction was true only
for some angles. The best and used today is law of refraction invented by Willebrord Snellius
in 1615. His law says that ratio sin(i) / sin(r) is constant for specified materials.
where:
i is the angle of incidence
r is angle of refraction
n1 is index of refraction of material 1
n2 is index of refraction of material 2
n12 is relative index of refraction of materials 1 and 2
This law is very useful because it allows us to work out how rays of light behave.
All we need to know is the index of refraction of first medium and the second one or the relative index of
refraction of this pair of materials. It is also used for identifying substances by their index
of refraction. There are some relative indexes of refraction of air and some other substances
in the Table 1.
Table 1
| Substance | Index of refraction |
| Glass | 1.5 - 1.9 |
| Diamond | 2.42 |
| Fused quartz | 1.46 |
| Quartz crystal | 1.54 |
| Glycerin | 1.47 |
| Carbon disulfide | 1.63 |
| Oleic acid | 1.46 |
| Water | 1.33 |
Total internal reflection
If we carry out an experiment on the light passing from a medium with a higher absolute
index of refraction to a medium with a lower index. For example, consider light passing
from glass with n=1.50, to air, at the angle of incidence of 0, 10, 20, 30, 40, 50 degrees.
The results will be as shown in Figures 1 - 6.
| 0 deg | 10 deg | 20 deg |
 |  |  |
| Figure 1 | Figure 2 | Figure 3 |
| 30 deg | 40 deg | 50 deg |
 |  |  |
| Figure 4 | Figure 5 | Figure 6 |
As you see in Figure 6, the incident ray, if passes at the particular
angle of 50 degrees, is not refracted, it is only reflected. Therefore, there must be
some angle of incidence between 40 deg and 50 deg at which refracted ray moves
parallel to the surface. The sine of the angle of incidence is given by Snell's law as:
sin(i)= sin(r) * n
But n is ratio of the sine of the incident ray to the sine of the refracted
ray of light as it passes from air into glass, so when it goes in opposite
direction the index of refraction is just the inverse of n. Therefore, when
light goes from glass into air the ratio of the sin(i) to the sin(r) is given as:
sin(i) / sin(r) = 1 / n
So the angle of incidence as light passes from glass to air is given as:
sin(i) = sin(r) / n
If we want the refracted ray to move parallel to the surface, the angle of
refraction must be 90 degrees (its sine is 1.00), so the sine of the angle
of incidence is:
sin(i)= 1 / n
Now, let's try to find the angle of incidence at which ray of light passing from
glass to air is refracted and moves parallel to the surface. Glass' index of refraction
is n=1.50, so the sine of the angle of incidence is:
sin(i)= 1 / 1.50 = 0.667
When sin(angle)=0.667, angle is 41.8 degrees; therefore the angle of incidence is 41.8 degrees.
As shown in the Figure 7, at this particular angle of incidence, the refracted beam moves parallel to the surface; it just glances
along the surface. This angle of refraction is the largest one possible. What
if a beam strikes the surface at an angle of incidence greater than 41.8 degrees?
This beam does not result in any refracted pencil; instead, it is totally reflected back
into the glass (as shown in Figure 6). The phenomenon is known as total internal reflection. The smallest angle
of incidence for which total internal reflection occurs is called the critical angle.
Figure 7 Here the angle of incidence is the critical angle (41.8 deg), so the refracted ray moves parallel to the surface
Frequently asked questions
Is the change from refraction to reflection rapid?
No, at small angles of incidence,
small part of light is reflected, almost all of it is refracted and at larger angles more and more light
is reflected and less is refracted.
What is the practical use of total internal reflection?
The phenomenon is used in light pipes. The light pipe consists of a cylindrical
rod of transparent plastic. Light enters at one end of the rod nearly normal to the
end surface. Any part of this light which reaches the side walls will have an incident
angle greater than the critical angle and therefore will not escape into the surrounding
air. Instead, a succession of total reflections will carry it along the rod and it will
finally emerge at the far end.
Refraction by prisms - dispersion
Thus far we have concentrated our attention on what happens to light
as it enters glass or some other substance from air or it leaves glass
into air. What happens when light goes from air into glass and then goes
from glass back into air? To find out what happens we can make use of
a glass block with parallel faces. Figure 8 shows the results of this
experiment.
Figure 8 A pencil of light
that is incident on one side of the glass leaves the glass on the other
side travelling parallel to the incident direction.
As you can see, a pencil of light that is incident on one side of the glass
leaves the glass on the other side travelling parallel to the incident direction.
We can, however change the direction of a light beam by using a piece of glass with
two nonparallel faces. Whenever the two faces are not parallel, the light
will emerge in a new direction. A piece of glass or plastic with two
nonparallel faces is called the prism.
Figure 9 Light bent by passing
through a glass block with non-parallel faces.
In theory, a prism should work as it is shown in Figure 9, but in reality
it does not. A careful examination shows that, even when the incident beam
is made of parallel pencils of light, the beam emerging from the prism
diverges, or spreads out (as shown in Figure 10). To investigate
this spreading, which does not seem to be quite consistent with the laws
of refraction, we shall allow the light to travel a considerable distance
from the prism and then examine it. We shall use a very narrow incident
pencil so that the spreding will be large compared to the pencil width.
A simple arrangement for doing this is shown in Figure 11. Light
from a distant source, such as the sun or an incandescent light bulb,
passes through a narrow aperture. This aperture produces a narrow beam
of light. It is found that the light falling on the screen is no longer
"white". Instead, a brilliant spectrum of colours is spread across the
screen. The colours are like those that we see in a rainbow, with red at
one end and violet at the other. The red deviates least from its original
direction, and the violet most. The spreading out of light into a spectrum is called dispersion.
Figure 10 A beam of white light diverging as a result of passing through a prism.
The divergence is exaggerated.
Figure 11 The dispersion of white light by a prism into a coloured spectrum.
Figure 12 A spectrum produced by dispersion of white light.
The deviation produced by a prism is determined by the angle between
the surfaces through which the light passes, by the direction of incidence
on the first face, and by the index of refraction of the prism. Of these,
the only quantity that can differ for the different colours of light is
the index of refraction. This leads us to the conclusion that the refractive
index depends on the colour of the light! Let's carry out an experiment to
check this theory. Figure 13 shows simple arrangement for doing this.
Two parallel beams of white light are "coloured" by set of two filters:
red and blue. Beams are then refracted by a prism and fall on the screen. It is found that
beams after leaving prism are no longer parallel, are wider than the incident ones and remain coloured,
they are not dispersed. This experiment leads
to another one conclusion: coloured light is not dispersed, it is only refracted.
Figure 13 Two parallel beams of different colours are no longer parallel
after refraction by a prism.
If this exlanation is correct, then we should expect to find the spreading
of white light into colours in some of our previous experiments on refraction.
Indeed, with careful examination of the refracted beam, it is possible
to see a slight separation of colours. But this effect is visible if we
examine the light at a considerable distance from the refraction point.
But the important thing is that the effect is present and seems to be
related to the nature of the refractive index, not some special property
of prism. The prism is simply a convenient shape for amplifying the effect into
something that is easily visible. Table 2 presents variation of
the index of refraction with colour for a glass that is used in many
lenses.
Table 2 Index of refraction of crown glass
| Colour | Violet | Blue | Green | Yellow | Orange | Red |
| Index | 1.532 | 1.528 | 1.519 | 1.517 | 1.514 | 1.513 |
Dispersion was first studied in the seventeenth century by Rene Descartes and
Sir Isaac Newton. Newton performed the additional experiment of trying to
break up one portion of the spectrum by inserting a prism in light of a
particular colour, say red. All that happens is that the red slightly further
spread out but it remains red. Unlike the original white light, it does not
split into a coloured spectrum. This leads to another conclusion: white light
is a mixture of lights of different colours, which are basic and cannot be
decomposed. Our conclusions are supported by a further experiment of Newton's
proving that white light is a combination of many colours. He recombined
the spectrum to make light white. We can do the experiment by breaking
up a narrow beam into a spectrum by a prism, and then placing in the spectrum
a second prism with greater angle between its faces (as shown in Figure 14).
Because of its greater angle this prism deviates the different colours more
than the original prism and the light converges again into the white light.
Figure 14 Light is first dispersed and than
combined again by set of prisms.
Lenses
The phenomenon of refraction of light has found usage in many devices.
Lenses are the most popular ones. Especially, cylindrical lenses.
Cylindrical lens is a piece of transparent material where the lines
representing the surfaces are arcs of circles or one is arc of circle and
the other is flat. The line passing through the center of the lens and on
which the centers of the two spheres are located is called the axis of lens.
The point on this axis at which incident parallel rays focus or converge
is the principal focus F. The distance of the principal focus from the
center of the lens is known as the focal length, f.
Figure 15 F - principal focus
f - focal length
The ray parallel to the axis is bent by the lens so as to pass through
the principal focus. It follows from the reversibility of light paths
that the ray that passes through the focal point must travel parallel
to the axis after it has passed through the lens.
Figure 16 Reversibility of light paths, rays sent from the
principal focus
travel parallel to the axis after they have passed through the lens.
Images formed by lenses
Lenses form real and virtual images. Real images are formed when the object
is located farther than the principal focal point. The real image can be made visible by placing
a screen on one side of the lens and the object on the other. Real images
are always upside down. If the object is far from the lens then the image
is close to the lens and is smaller than the object, if the object is located
near the lens then the image is formed far from the lens and is bigger than the object.
Figure 17 The real image of candle is formed on the screen. Its size depends on the distance of the object from the lens.
Figure 18 The real image, upside down, smaller than the object.
Figure 19 The real image, upside down, of original size.
Figure 20 The real image, upside down, bigger than the object.
Virtual images are formed when the object is placed between the principal
focal and the lens. You can see it by looking straight at the lens.
Figure 21 The virtual image, straight, bigger than the object.
