## A Brief Look at Mirages – Li Kewei

From grim tales of that delusional oasis to legends of that mysterious ship in the sky, mirages have long captured the imagination and fascination of men. They are definitely amongst the most famous and fascinating of atmospheric optical phenomena. The word mirage came from the Latin mirare, meaning “to look at, to wonder at”. In this FOTW, we will explore the physics behind mirages, in all its hallucinating variants.

fig 1: Picture of a fictitious "oasis". Note that the blue colour is below the horizon, giving the impression of a distant water source.

Mirages are formed by refraction, which requires a varying refractive index. For the sake of formality, I’ll remind the reader what the refractive index of a material is:

$n=\frac{c}{v}$

where c is the speed of light in vacuum and v is the (phase) velocity of light in that material. The greater the refractive index, the slower light travels through the material.

All mirages are formed because the refractive index of air varies with temperature. The higher the temperature, the lower the refractive index is. This is intuitively correct, as higher temperatures lead to reduced density air, and thus reduced optically density too. However it is to be noted that although density and optical density are related in the case of air, they are two distinct concepts in most other physical situations. Now let us find a more quantitative relationship between the refractive index and temperature.

Another way to define the refractive index is , where  and  are the relative permittivity and permeability of the material respectively. In air,   and variations in  accounts for nearly all the changes in n. The refractive index of air is very close, but not exactly equal, to 1. In fact . This value, however, is not constant, but varies with temperature. From theoretical analysis, it is known that

Where N is the number density of air particles and  is the molecular polarizability. As a consequence of the gas laws,  , where T is the absolute temperature. If we model air as an ideal gas, it’s not hard to arrive at the result that:

where  is the atmospheric pressure and  is the Boltzmann’s constant. From this equation we can see that the refractive index decreases as temperature increases, as mentioned before.

Fig 2: Illustration of a typical mirage.

Now let us move on explore how mirages actually form. Figure 2 shows the typical mirage that is often associated with the apocryphal oasis. In this situation, we have a hot pavement that is heated up by the sun, resulting in a vertical temperature gradient, and thus a refractive index gradient. Light from the sky undergoes a series of infinitesimal refractions before entering the eyes of the observer from below, thus appearing to have originated from somewhere on the ground in front of the observer. The fictitious oasis seen by the observer is actually blue light from the sky.

One way to think about the situation is to apply Snell’s Law to every infinitesimal refraction. Since , we observe that  is a conserved quantity, allowing us to do some mathematics. However, that is rather tedious and doesn’t give a good intuitive feel of why the light bends. A more intuitive way to make sense of the situation is to apply *Fermat’s principle of least time. Suppose our light ray starts in the sky and ends up in the observer’s eye. What is the path of least time given these starting and ending points? If the refractive index were uniform, then it would just be a straight line. However, now the air at the bottom is hotter, thus light could travel faster the nearer it went to the ground. It makes sense, then, for the light to travel closer to the ground for a while. It cannot overdo this though, since going too far down would make the path too long. If you work out the maths, the optimal balance would arise when light follows Snell’s law, as expected.

Normal mirages that form over the surface of a hot road or desert can be called inferior mirages, because the light appears to emerge from below the horizon. On the other hand, superior mirages are formed over a surface that is colder than the surround air,

Fig 3: A superior mirage.

Fig 4: The Flying Dutchman?

such as over a chilly ocean. This gave rise to images of ships floating in mid air, albeit upside down (figure 3). This appears to be rather well documented (figure 4), and has even been proposed to be the origins of the Flying Dutchman.

A even more mysterious form of superior mirage is known as the Fata Morgana. The name is an Italian phrase derived from the vulgar Latin for “fairy” and the Arthurian sorcerer Morgan le Fay, from a belief that the mirage, often seen in theStraitofMessina, were fairy castles in the air or false land designed to lure sailors to their death created by her witchcraft. Fata Morgana mirages arise due to different layers of air having different temperatures, and are thus extremely complex. The mirage typically distorts the objects on which it is based, often to the extent that the original objects become unrecognizable. Because of it depends on a delicate multilayered atmospheric condition, the mirage is often rapidly changing too. A video of a Fata Morgana based on a boat can be found here:

http://en.wikipedia.org/wiki/File:Fata_Morgana_is_changing_
shape_of_a_distant_boat_.OGG

Note:

*Fermat’s Principle of least time: It states that the path taken between two points by a ray of light is the path that can be traversed in the extreme value of time (meaning least, most, and stationary value, corresponding to zero-valued first derivative in mathematics). Fermat’s Principal of Least Time, together with Lagrange’s variational priciple, provides a basis for the study of the geometry of space and it is widely used in General Theory of Relativity. (Added by Qiushi when editing)