Fundamentals of geometrical optics

Refraction laws

Let us consider surface S in a space, separating two medium of indexes n 1 and n2 respectively containing points A and B. The path effectively taken by light to go from A to B runs through Point A on the surface.

AI is the the incident ray, IB is the refracted ray.

L = n1.AI + n2.IB is the optical path (AIB).

Slight displacement dI of I causes a variation dL such that dL/dI = 0 according to Fermat's principle. The expression (3) applied to the trajectory AI and IB give us :

Similarly :

Since , finally we get :

If is the unitary vector of the normal, is that in the direction that is :

For a trajectory effectively followed by light, Fermat's principle imposing, dL/dI = 0 for every , and are perpendicular, therefore and are parallel.

shows that and belong to the same plane P.

P is the incidence plane, it contains the incident ray, the refracted ray and the normal to the surface in I, i1 and i2 are, in this plane, the angles between the incident and refracted rays in relation to the normal. We deduct the vectorial refraction relationship :

And by refraction in the surface plane :

Descartes' laws are deducted from the previous relationships :

Law 1 : The refracted ray is in the incidence plan

Law 3 : The angles i1 et i2 of incident and reflected rays are such that n1.sini1 = n2.sini2

La loi 2 is concerned with reflecting surfaces, for which i1 = - i2 . We will see later that the formulas for refracting surfaces can be applied to reflecting surfaces with: n2=-n1.

Figure 05
Figure 05 [zoom...]

Refraction

Let us consider two medium of indexes n1 et n2, Descartes' third law gives us the relationship between the incidence and refraction angles in both mediums :

This relationship is perfectly symmetrical, in accordance to the principle of reversibility of light. A ray arising from the second medium in an angle i2 with the normal will form, after refraction from the surface, an angle i1 in the first medium which meets with the relationship (5).

Figure 06
Figure 06 [zoom...]

Limit angle

Limit angle

For il = 90°, a grazing incidence angle, the refracted ray forms, with the normal, the angle i2 such that : sini2=n1/n2.

i 2 is thus called the limit angle il .

Any ray arising from the second medium of which the incidence angle i2 is higher than il will undergo total reflexion.  Si le premier milieu est l'air (n1=1), le tableau ci-dessous donne quelques valeurs de il : PAS TRADUIT

[zoom...]
Figure 07 & Figure 08
Figure 07 & Figure 08 [zoom...]

Total reflection in a multimodal optical fiber

Figure 09
Figure 09 [zoom...]

The core index is n1, the clad index is n2< n1.

A light ray with an incidence over i1 on the core-clad interface reflects itself entirely, it is guided.

Its incidence at fiber entry is lower than .

is the half angle aperture of the fiber. The digital opening of the fiber is :

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