Chapter 9

Geometrical and waves optics

Interpretation of mirages

Take 15 minutes to prepare this exercise.

Then, if you lack ideas to begin, look at the given clue and start searching for the solution.

A detailed solution is then proposed to you.

If you have more questions, feel free to ask them on the forum.

We study the propagation of a light beam in a vertical plane (Oxz) where the index of the medium (air) varies according to the law :

Where and are constant.

Interpretation of mirages
Comment courber un rayon lumineux ?

Question

Show that :

Hint

Apply Descartes relationship for refraction, valid here for a stratified continuous medium.

Solution

The index equal lines are given by (these are horizontal planes).

The law of refraction of Descartes writes :

So :

Question

Determine the differential equation of the trajectory of the light beam :

Hint

Use the fact that

Solution

We can write the relationship between the angle and the elementary variations and of geometric coordinates of the light beam on its path :

Therefore :

Question

Show that the beam path is a parabola.

Hint

Think derive the previous differential equation over .

Solution

Using the expression of , the above differential equation becomes :

This equation is derived in relation to  :

Therefore :

This equation is solved by :

Where we used the boundary conditions :

and

The trajectory of the light beam in the in-homogeneous medium is a parabola.

Question

Show that this theory applies to mirages.

We will justify the change in the index of a gas with altitude from the relationship of Gladstone :

Where is the density of the gas. We can assume that the temperature varies with altitude as follows :

Where is a constant (and we can assume that ).

Hint

Determine the expression of using the fundamental relationship of fluid statics : .

Solution

The gas is air, likened to an ideal gas. The equation of state can be written as :

Where is the molar mass of air.

Moreover, the basic relationship of the hydrostatic gives :

Then obtained a differential equation satisfied by the pressure :

So :

We deduce by integration :

As , one can perform the limited development :

Gladstone's law is written :

So :

By using the obtained relationship between pressure and temperature, we get :

Development in the first order :

Hence the expression :

Can thus be identified :

And :

Finally :

  • Inferior Mirage (or hot mirage) :

    A ground plane, arid, overheated by the sun creates temperature conditions given above.

    Bitumen of a road creates the same phenomenon.

    Light rays become parabolic and give the impression to be reflected, analogous to the reflection on a mirror.

Hot mirage
  • Superior Mirage (or cold mirage) :

    A reverse phenomenon can be created when atmospheric layers are warmer and the temperature gradient decreases with altitude drop : it is a phenomenon typically observed at sea.

Cold mirage
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Fermat's principle
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Scalar theory of light, light intensity (illumination)