Waves at an interface

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The main thread of this web page deals with the behaviour of a ray of light (reflection, refraction and dispersion) when it meets an air/glass boundary. But many of the ideas apply just as well to other situations. The aim of this page is to enable you correctly to sketch all the rays that exist.

It is crucial that you appreciate the importance of the normal

At the point where the ray of light meets the boundary, we draw a normal, which is a line perpendicular to the boundary, and extending both sides of the boundary (so that the boundary and normal, taken together, make up a plus sign, as indicated by the thin black cross on the diagram). The angle associated with a ray is the angle between the ray and the nearest branch of the normal, rather than the perhaps more obvious angle between the ray and the boundary. In this diagram the angle has been labelled i, standing for 'angle of incidence'. The incoming ray is called the 'incident ray'.

Notice that the little black cross (which you do not usually draw, by the way - just the normal will do) divides the paper up into four quadrants. One of those quadrants is already filled by the incident ray. The question is: which of the other three quadrants have rays in them, and which way do the arrows point ?

If the light stays in the same medium (air or glass), we have reflection. If it enters the other medium, we have refraction.

Rule 1: Whether we get reflection or refraction, the ray always crosses over to the other side of the normal.

Reflection

Rule 2: We always get reflection, but the reflection may be total or partial, depending on circumstances.

To make the point that there are many possibilities, this time the incident ray is shown as starting in the glass, and it travels basically from right to left instead of the other way round. But the law of reflection is still obeyed: The angle of incidence is equal to the angle of reflection. This law is obeyed for all colours, so there is just one angle of reflection for a white ray (which is, of course, composed of many colours). Notice that the arrows could perfectly well be the other way round. When one medium is surrounded by the other as, for example, in the case of a glass block surrounded by air, if the reflection occurs in the 'inner' (surrounded) medium then it is called internal reflection (as depicted here).

Refraction

We will refer to the medium in which waves travel more quickly (the air) as the 'fast' medium and to the medium in which the waves travel more slowly (the glass) as the 'slow' medium. Note the use of the quotes: the air and glass are not themselves moving!

Although the speed of light in air is virtually the same for all colours, it differs slightly for the different colours in glass. But, for any given colour the speed has a single value in a given medium, irrespective of the direction in which the light is travelling. So, for the time being, we must restrict ourselves to light of one colour. Let's choose green. Green light has one speed in air (3´10⁸ m/s ) and another in glass (about 2´10⁸ m/s). The ratio of these two speeds is called the refractive index, denoted by n, and a simple calculation (dividing one by the other) shows that the ratio has the value 1.5 for green light in glass. So we have

or, in symbols

We have seen (Rule 2) that light encountering a boundary always reflects. Sometimes, part of the ray refracts as well - it will do this if it is able to move from one medium to the other. Concentrate, for now, on what happens to rays that do refract. The next diagram shows just the refraction, but bear in mind that a reflected ray always exists as well.

Notice several features of this diagram: We are dealing with monochromatic light (one colour only). The rays have no arrows: you do not have to worry about which way the light is moving. The slowest moving ray of light makes the smallest angle with the normal. The ray crosses both the boundary and the normal as it goes from one medium into the other.

Rule 3: Refraction occurs in such a way that the angle in the 'slow' medium is smaller than the angle in the 'fast' medium - but it will only occur at all if Snell's Law leads to the 'small' angle being less than the critical angle. Refraction can only ever be partial, since Rule 2 tells us that some of the wave is always reflected. Note that refraction will always occur whenever the incident ray is in the 'fast' medium.

Rule 1 (again): All 'going away' rays must end up on opposite side of the normal to the incident ray. You cannot use all four quadrants of the plus sign without violating this rule!

Putting it all together

Let us now draw some diagrams that incorporate both reflection and refraction, and which include the direction arrows. To understand this part of the page, you need to know what is meant by the critical angle. For glass in air it has a value of 42^o.

The incident ray (the solid line) is partially reflected, as shown by the dotted line. The incident ray is in the 'fast' medium, so refraction occurs in accordance with Rule 3, the angle in the 'slow' medium being the 'small' angle, and hence less than 60o. Click here for details of the calculation. When the refracted ray, shown by the dashed line, strikes the second boundary, it is reflected at an equal angle in accordance with Rule 2. Notice that we need a new normal in order to be able to deal with this second boundary. The 'small' angle (in the 'slow' medium) at this second boundary is greater than the critical angle, so, by Rule 3, there is no refraction. Thus the reflection that occurs is total internal reflection.

Here is magenta light going into a prism. Note that the light blue colour of the prism is only there in an attempt to make the prism stand out from the air. Both are in reality colourless. This arrangement splits light up into its colours, forming a spectrum. The splitting up is called dispersion.

For the sake of clarity, the reflected rays have been omitted from this diagram. Click here to see a diagram that includes the reflections. The following explanation will require careful reading. Magenta light contains a red component and a blue component. These travel at different speeds in glass, but at the same speed in air. Therefore, for a given 'big' angle (the angle the magenta ray makes with the first normal) they have different 'small' angles in the glass. So the red and blue rays set off in different directions in the glass, the red 'small' angle being bigger than the blue 'small' angle: i.e. there is a larger difference between 'big' and 'small' for blue than there is for red. At the second face the blue 'small' angle is bigger than the red' small angle, but both are less than their respective critical values, so both are able to refract out. We noted above that the difference between 'big' and 'small' for blue is larger than the difference for red. So, since small blue is already larger than small red, big blue is quite a lot larger than big red, and the two rays diverge, as shown.

Tony Ayres, 13 February 2005