What is the refractive index

• Refraction is a deviation in the path of waves as they pass from one medium to another. The change in path is produced by the change in the propagation speed of the waves in the two media (see refraction). The refractive index is a quantity that indicates how much a medium changes the speed of a certain type of wave. In its most common sense the term refers to the refraction of electromagnetic waves, and in particular of light.
• Naturally, the refractive index depends on the nature of the waves (e.g. every medium has an optical refractive index and an acoustic one, which refer respectively to electromagnetic waves and to sound waves). A medium can have different refractive indices even for waves of the same type but with different parameters. For example, the optical refractive index generally depends on the wavelength of the waves, while the refractive index for longitudinal elastic waves (such as sound waves) is generally different from that for transverse elastic waves, etc.
• Physical investigation must lead us to discover several properties of the refractive index, and in particular:
  1. the connection between the value of the refractive index and the path of the waves
  2. the connection between the value of the refractive index and the fundamental properties of the medium (such as its density, its temperature and, more deeply, its microscopic structure).
• Let us begin by answering the first question in the simple case of a wave (e.g. a light wave) crossing the surface that separates two different semi-infinite media.

Refractive index and path of the waves

The law governing the relation between the direction of the incident ray and that of the refracted ray was found experimentally, in the case of light, by the Dutchman Willebrord Snell (1580-1626). Calling ${\displaystyle {\theta }_i,{\theta }_r}$ respectively the angle of incidence and the angle of refraction, Snell's law says that ${\displaystyle \frac{\sin {\theta }_i}{\sin {\theta }_r}=n_{12}}$ where n12 is a constant called the relative refractive index of medium 2 with respect to medium 1. It is tabulated for the various substances (at a precise wavelength) taking as medium 1 the vacuum, which is conventionally assigned a refractive index equal to one. The refractive indices tabulated in this way are called absolute refractive indices. They are pure numbers, always greater than 1.

Absolute refractive indices make it possible to obtain relative refractive indices according to the rule

${\displaystyle n_{12}=\frac{n_2}{n_1}}$

(1)

whose explanation will become immediate as soon as the physical meaning of the refractive index is understood. (1) then provides a kind of multiplication rule for refractive indices, useful when multiple refractions have to be analysed

${\displaystyle n_{13}=n_{12}\cdot n_{23}}$

(2)

Using absolute refractive indices, Snell's law takes the form

${\displaystyle \frac{\sin {\theta }_i}{\sin {\theta }_r}=\frac{n_2}{n_1}}$

Note that in this form the law gives an immediate mathematical consequence: if

${\displaystyle n_2>n_1\to {\theta }_i>{\theta }_r}$

(3)

This allows us to predict that

• in passing from a medium with a higher refractive index to a medium with a lower refractive index, the refracted ray moves away from the normal;
• in passing from a medium with a lower refractive index to a medium with a higher refractive index, the refracted ray moves towards the normal;

Corpuscular and wave interpretation of refraction

The phenomenon of refraction is interesting because it could have provided (had the experimental means of the late 17th century allowed it) an experimentum crucis to settle the question between the corpuscular theory and the wave theory of light well before Young's experiment.

Corpuscular explanation

According to the corpuscular theory of light, the refractive index is closely connected to the density of the material (denser materials generally have a higher refractive index). The ray, in passing from a less dense medium to a denser medium, moves towards the normal because of the greater gravitational attraction that the denser medium exerts on the light corpuscles. The size of this deviation depends on the different densities of the media. This difference is "measured" by the refractive index, which takes ad hoc values to explain the observed deviations. To understand what happens we can use a gravitational analogue: imagine looking from above at a ball that, moving on a horizontal plane, meets a downward slope. If the ball enters the slope with a velocity at a certain angle to the plane-slope boundary, it changes direction because the component of the velocity perpendicular to the boundary increases because of the acceleration of gravity. The interesting thing is that, according to the corpuscular interpretation, light travels faster in media with a higher refractive index (denser, according to Newton)

Wave explanation

Using Huygens' principle, the wave theory explains refraction by assuming that light travels more slowly in media with a higher refractive index. It is then clear what was meant by the initial remark: by being able to measure the speed of light in a medium (a measurement first made by Foucault in 1850), a decisive proof in favour of the wave theory of light would have been obtained already at the end of the 17th century. The superiority of the wave interpretation does not end there: it provides a physical interpretation of the refractive index that immediately justifies the multiplication rule (2). The refractive index gives the ratio between the speeds of the wave in the two media. In particular

${\displaystyle n_{12}=\frac{v_1}{v_2}}$

longitudinal waves cross regions of different refractive index	illustration of the phenomenon by means of Huygens' principle.
It is evident that the waves in the two media travel together at the interface. However, the propagation speed is greater in the upper medium than in the lower one, and consequently, precisely in order to maintain continuity at the interface, the wavelength must become shorter, which produces the change in direction.	If every point at the interface becomes a source of spherical waves, all the point sources produce constructive interference along particular lines (in this case straight lines), which form the wavefronts of the refracted waves. For convenience, the normal to the fronts has also been marked in the figure; in the geometrical picture it corresponds to the wave ray

The incident wave (whose wavefronts we have drawn one wavelength ${\displaystyle \lambda }$ apart) advances with speed

${\displaystyle v_1=\frac{\lambda }{T}}$

(4)

and strikes the separating surface first at point ${\displaystyle S_1}$, then, after one period T, at point ${\displaystyle S_2}$ and, after another period T, at point ${\displaystyle S_3}$. If, according to Huygens' principle, we assume that these points, at the moment they are reached by the wavefronts of the incident wave, become in turn secondary sources propagating in the denser medium at a lower speed

${\displaystyle v_2=\frac{{\lambda }^{\prime }}{T}}$

(5)

it is easy to see that the wavefront of the secondary waves, that is, the envelope given by the segment ${\displaystyle \overline{A{S}_3}}$, changes direction with respect to the wavefront of the incident wave. In particular, applying the definition of sine to the right triangle ${\displaystyle A{S}_3{S}_2}$ we obtain:

${\displaystyle \sin {\theta }_r=\frac{{\lambda }^{\prime }}{\overline{S_2{S}_3}}}$

from which, since

${\displaystyle \overline{S_2{S}_3}=\frac{\lambda }{\sin {\theta }_i}}$

we obtain with simple algebra, also using (4) and (5),

${\displaystyle \frac{\sin {\theta }_r}{\sin {\theta }_i}=\frac{v_2}{v_1}}$.

(6)

Comparing (6) with Snell's law we obtain an immediate interpretation of the relative refractive index ${\displaystyle n_{12}}$ of medium 2 with respect to medium 1

${\displaystyle n_{12}=\frac{v_1}{v_2}}$

or in terms of absolute refractive indices

${\displaystyle \frac{n_1}{n_2}=\frac{v_2}{v_1}}$

(7)

If in particular we choose the vacuum as medium 1 (with refractive index n=1), (7) takes the form

${\displaystyle v_2=\frac{v_{\mathrm{v}\mathrm{a}\mathrm{c}\mathrm{u}\mathrm{u}\mathrm{m}}}{n_2}=\frac{c}{n_2}}$,

(8)

where, as usual, c≈300000 km/s denotes the speed of light in vacuum.

In summary, Huygens' principle, with the sole assumption that the wave travels more slowly in the medium with the higher refractive index, is able to:

• justify the change in the direction of propagation of the refracted ray;
• provide a physical interpretation of the absolute refractive index of a medium. This pure number, by (8), gives the ratio between the speed of the wave in vacuum and the speed of the wave in the medium. It is always greater than 1 because light travels more slowly in the medium than in vacuum.
• show why the multiplication rule works

${\displaystyle n_{13}=n_{12}\cdot n_{23}}$.

By (7), in fact,

${\displaystyle n_{13}=\frac{v_1}{v_3}=\frac{v_1}{v_2}\cdot \frac{v_2}{v_3}=n_{12}\cdot n_{23}}$

• explain why, in passing from one medium to another, it is the wavelength and not the frequency of the wave that changes. This is tacitly assumed in the previous proof when we required that the secondary waves emit with the same period (and therefore frequency) as the primary (incident) wave

Phenomena due to refraction

Total reflection

We have said that in passing from a denser medium to a less dense medium (which for simplicity we will assume to be the vacuum) the ray moves away from the normal. What happens if the angle of incidence of the ray is such that, by Snell's law, the angle of refraction becomes equal to 90°? This happens at a precise angle θlim, called the critical angle, which can be derived from Snell's law (as usual, using index 1 for the "starting" medium and index 2 for the destination medium, that is, the vacuum) ${\displaystyle \frac{\sin {\theta }_{\mathrm{l}\mathrm{i}\mathrm{m}}}{\sin 90^{\circ }}=\frac{1}{n_1}\to {\theta }_{\mathrm{l}\mathrm{i}\mathrm{m}}=\arcsin \frac{1}{n_1}}$

The refracted ray becomes "grazing" to the separating surface and cannot leave the denser medium. The deviation away from the normal becomes more marked for angles of incidence greater than the critical angle: the refracted ray (which now no longer "deserves" that name) is actually completely reflected. This is the phenomenon of total (or internal) reflection

Applications of total reflection: optical fibres

The phenomenon of total reflection has an important application in the field of optical fibres. Optical fibres are waveguides for light. They consist of a tube of transparent material (such as glass, or certain polymers), surrounded by a cladding of a material with a lower refractive index. If a ray of light sent into the fibre is made to strike the surface at a very grazing angle (above the critical angle), it undergoes total reflection many times and remains trapped in the fibre. It can leave the fibre, without being refracted outside it, only where the fibre ends. Clicking on the image alongside shows its high-resolution version, in which one can clearly see the points where the laser beam touches the inner wall of the fibre and undergoes total reflection. The laser is the thin red line that propagates in a straight line inside the fibre until, on contact with one of the walls, it undergoes multiple total reflections.

Mirages and other strange phenomena

• Have you ever noticed that an object lying on the bottom of a swimming pool appears, to an outside observer, in an "apparent" position different from the real one? Or that a stick partly immersed in water appears "broken"?
• Have you ever wondered why, on hot summer days, the asphalt of a long straight road looks almost "wet", as if it were the surface of a lake?
• Have you ever heard of the Fata Morgana phenomenon?

These phenomena can be explained by refraction and, in the last two cases, by the phenomenon of total reflection.

• When an object, for example a coin, lies on the bottom of a swimming pool, the rays of light reflected from it, on leaving the water, are bent by refraction away from the normal to the separating surface. An outside observer hit by these refracted rays, used to thinking that light travels in a straight line, will assign to the object a position on the bottom different from the real one. The same happens in the atmosphere, but temperature gradients and convection currents in the air can make the apparent image rather distorted. For example, it is well known that, because of atmospheric refraction, the setting sun remains visible above the horizon longer than it would without an atmosphere. In the illustration alongside, besides this effect, one can also see a curious false image produced by non-uniform refraction in the lowest layers of air.

• When in summer the asphalt, or the sand, reaches very high temperatures, the layer of air immediately above it becomes very hot and therefore made of more rarefied air. The layer immediately above will be less hot, so the air denser, and so on. Since the refractive index is closely linked to the density (and therefore to the temperature) of the air, we can imagine that it varies continuously in passing from a zone of higher to a zone of lower density. If a wave (for example a ray of sunlight) moves through a medium that we can imagine, for simplicity, as made of "layers" of decreasing refractive index, its path, passing from one layer to the next, bends away from the normal. At a certain point the angle of incidence of the ray exceeds the critical angle: the ray is completely reflected towards the upper layers. An observer hit by the ray of sunlight will think that it comes from below (from the asphalt) and will see a "virtual" sun in the asphalt, which they interpret as a kind of puddle. Obviously the conditions that make this phenomenon, called a mirage, possible are due to the strong temperature gradient between one layer of the atmosphere and the next. In the scorching desert these conditions are obviously met as well!

Under certain environmental conditions an even more surprising phenomenon can occur, in some ways opposite to the one described: if the air is made of layers of increasing refractive index, the rays coming from sources at ground level will seem to come from sources raised above the ground (the classic example quoted in these cases is the oasis in the desert that appears high above the horizon, often with an upside-down image). The two cases are sometimes distinguished by speaking of an inferior mirage and a superior mirage

inferior mirage	superior mirage

• The Fata Morgana phenomenon is a particular type of mirage in which the apparent images are produced at a certain height above the ground, and change rapidly, just like the apparitions of the character of the same name in Celtic mythology. The explanation of the phenomenon is similar to that of the mirage, but in this case the refractive index between the various layers of air varies much more irregularly: up to a certain height it increases, and then decreases again. Moreover, it changes rapidly in time: this is why, unlike the mirage, the images produced by the effect are very changeable and deformed, hard to recognise. The images produced often look like towers and obelisks, giving rise to a kind of unreal city that transforms and vanishes in a very short time. A single image therefore cannot convey how effective the illusion is, but can only show the vertical structure of the apparent image.

A gallery of images of various mirages with comments can also be seen at http://www.meniero.it/articoli/meteo/it/Introduzione%20alle%20fotometeore.html.