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
SubstanceIndex of refraction
Glass1.5 - 1.9
Diamond2.42
Fused quartz1.46
Quartz crystal1.54
Glycerin1.47
Carbon disulfide1.63
Oleic acid1.46
Water1.33

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 deg10 deg20 deg
Figure 1Figure 2Figure 3

30 deg40 deg50 deg
Figure 4Figure 5Figure 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

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
ColourVioletBlueGreenYellowOrangeRed
Index1.5321.5281.5191.5171.5141.513

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.