# Fresnel's Formula

##### 6 minutes • 1237 words

Fresnel’s second supplement to his first memoir on Double Refraction was presented to the Academy on November 26, 1821[62].

It indicated the lines on which his theory might be extended so as to take account of dispersion:

“The molecular groups, or the particles of bodies may be separated by intervals which, though small, are certainly not altogether insensible relatively to the length of a wave.”

Such a coarse-grainedness of the medium would, as he foresaw, introduce into the equations terms by which dispersion might be explained.

The theory of dispersion which was afterwards given by Cauchy was actually based on this principle. It seems likely that, towards the close of his life, Fresnel was contemplating a great memoir on dispersion,[63] which was never completed.

Fresnel had reason at first to be pleased with the reception of his work on the optics of crystals: for in August, 1822, Laplace spoke highly of it in public.

At the end of the year a seat in the Academy became vacant, he was encouraged to hope that the choice would fall on him.

In this he was disappointed.[64]

Meanwhile, his researches were steadily continued. In January, 1823, the very month of his rejection, he presented to the Academy a theory in which reflexion and refraction[65] are referred to the dynamical properties of the luminiferous media.

As in his previous investigations, he assumes that the vibrations which constitute light are executed at right angles to the plane of polarization.

He adopts Young’s principle, that reflexion and refraction are due to differences in the inertia of the aether in different material bodies, and supposes (as in his memoir on Aberration) that the inertia is proportional to the inverse square of the velocity of propagation of light in the medium.

The conditions which he proposes to satisfy at the interface between two media are that the displacements of the adjacent molecules, resolved parallel to this interface, shall be equal in the two media, and that the energy of the reflected and refracted waves together shall be equal to that of the incident wave.

On these assumptions the intensity of the reflected and refracted light may be obtained in the following way:—

Consider first the case in which the incident light is polarized in the plane of incidence, so that the displacement is at right angles to the plane of incidence; let the amplitude of the displacement at a given point of the interface be f for the incident ray, g for the reflected ray, and h for the refracted ray.

The quantities of energy propagated per second across unit cross-section of the incident, reflected, and refracted beams are proportional respectively to

where c1, c2, denote the velocities of light, and ρ1,ρ2 the densities of aether, in the two media, and the cross-sections of the beams which meet the interface in unit area are

cos i, cos i, cor r

respectively. The principle of conservation of energy therefore gives

The equation of continuity of displacement at the interface is

```
f + g = h.
```

Eliminating h between these two equations, and using the formulae

we obtain the equation

Thus when the light is polarized in the plane of reflexion, the amplitude of the reflected wave is

the amplitude of the incident vibration.

Fresnel shows in a similar way that when the light is polarized at right angles to the plane of reflexion, the ratio of the amplitudes of the reflected and incident waves is

These formulæ are generally known as Fresnel’s sine-law and Fresnel’s tangent-law respectively. They had, however, been discovered experimentally by Brewster some years previously. When the incidence is perpendicular, so that i and r are very small, the ratio of the amplitudes becomes , where μ2 and μ1 denote the refractive indices of the media. ‘This formula had been given previously by Young[66] and Poisson,[67] on the supposition that the elasticity of the aether is of the same kind as that of air in sound.

When i + r = 90°, tan (i + r) becomes infinite: and thus a theoretical explanation is obtained for Brewster’s law, that if the incidence is such as to make the reflected and refracted rays perpendicular to each other, the reflected light will he wholly polarized in the plane of reflexion.

Fresnel’s investigation can scarcely be called a dynamical theory in the strict sense, as the qualities of the medium are not defined. His method was to work backwards from the known properties of light, in the hope of arriving at a mechanism to which they could be attributed; he succeeded in accounting for the phenomena in terms of a few simple principles, but was not able to specify an aether which would in turn account for these principles. The “displacement” of Fresnel could not be a displacement in an elastic solid of the usual type, since its normal component is not continuous across the interface between two media.[68]

The theory of ordinary reflexion was completed by a discussion of the case in which light is reflected totally. This had formed the subject of some of Fresnel’s experimental researches several years before .

In 2 papers[69] presented to the Academy in November, 1817, and January, 1818, he had shown that light polarized in any plane inclined to the plane of reflexion is partly “depolarized” by total reflexion.

This is due to differences of phase which are introduced between the components polarized in and perpendicular to the plane of reflexion.

“When the reflexion is total,” he said, “rays polarized in the plane of reflexion are reflected nearer the surface of the glass than those polarized at right angles to the same plane, so that there is a difference in the paths described.”

This change of phase he now deduced from the formulae already obtained for ordinary reflexion. Considering light polarized in the plane of reflexion, the ratio of the amplitudes of the reflected and incident light is, as we have seen,

when the sine of the angle of incidence is greater than μ2/μ1, so that total reflexion takes place, this ratio may be written in the form

where θ denotes a real quantity defined by the equation

Fresnel interpreted this expression to mean that the amplitude of the reflected light is equal to that of the incident, but that the two waves differ in phase by an amount θ. The case of light polarized at right angles to the plane of reflexion may be treated in the same way, and the resulting formulae are completely confirmed by experiment.

A few months after the memoir on reflexion had been presented, Fresnel was elected to a seat in the Academy, and during the rest of his short life honours came to him both from France and abroad.

In 1827, the Royal Society awarded him the Rumford medal; but Arago, to whom Young had confided the mission of conveying the medal, found him dying; and eight days afterwards he breathed his last.

By the genius of Young and Fresnel the wave-theory of light was established in a position which has since remained unquestioned. It seemed almost a work of supererogation when, in 1850, Foucault[70] and Fizeau,[71] carrying out a plan long before imagined by Arago, directly measured the velocity of light in air and in water, and found that on the question so long debated between the rival schools the adherents of the undulatory theory had been in the right.